Authors Malloy • Molix-Bailey Price • Willard
(bkdg)Created by Michael Trott with Mathematica. From Graphica 1, Copyright ©1999 Wolfram Media, Inc., (b)Richard Cummins/SuperStock
About the Cover The Space Needle, designed as a symbol of the 1962 World’s Fair, is now the most popular tourist destination in Seattle, Washington. The Needle is 605 feet tall, which made it the tallest building west of the Mississippi River when it was built. The Needle can withstand a wind velocity of 200 miles per hour. It has also withstood several earthquakes, including one in 2001 that measured 6.8 on the Richter scale. You’ll learn more about how mathematics and architecture are related in Chapters 10 and 11. About the Graphics Created with Mathematica. A rectangular array of circles is progressively enlarged and rotated randomly. For more information, and for programs to construct such graphics, see: www.wolfram.com.
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Microsoft ® Excel® is a registered trademark of Microsoft Corporation in the United States and other countries. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-873818-0 MHID: 0-07-873818-0 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 043/071 16 15 14 13 12 11 10 09 08 07
C
e s in
r ef
Start Smart Unit 1 Algebra and Integers 1
The Tools of Algebra
2
Integers
3
Equations
Unit 2 Algebra and Rational Numbers 4
Factors and Fractions
5
Rational Numbers
Unit 3 Linear Equations, Inequalities, and Functions 6
Ratio, Proportion, and Percent
7
Functions and Graphing
8
Equations and Inequalities
Unit 4 Applying Algebra to Geometry 9
Real Numbers and Right Triangles
10
Two-Dimensional Figures
11
Three-Dimensional Figures
Unit 5 Extending Algebra to Statistics and Polynomials 12
More Statistics and Probability
13
Polynomials and Nonlinear Functions
iii
Authors
Carol E. Malloy, Ph.D. Associate Professor University of North Carolina at Chapel Hill Chapel Hill, North Carolina
Rhonda J. Molix-Bailey Mathematics Consultant Mathematics by Design DeSoto, Texas
Jack Price, Ed.D. Professor Emeritus California State Polytechnic University Pomona, California
Teri Willard, Ed.D. Assistant Professor Department of Mathematics Central Washington University Ellensburg, Washington
Contributing Author Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas
iv Aaron Haupt
Meet the Authors at pre-alg.com
Consultants Glencoe/McGraw-Hill wishes to thank the following professionals for their feedback. They were instrumental in providing valuable input toward the development of this program.
Differentiated Instruction
Gifted and Talented
Mathematical Fluency
Nancy Frey, Ph.D. Associate Professor of Literacy San Diego State University San Diego, California
Ed Zaccaro Author Mathematics and science books for gifted children Bellevue, Iowa
Jason Mutford Mathematics Instructor Coxsackie-Athens Central School District Coxsackie, New York
English Language Learners Mary Avalos, Ph.D. Assistant Chair, Teaching and Learning Assistant Research Professor University of Miami, School of Education Coral Gables, Florida Jana Echevarria, Ph.D. Professor, College of Education California State University, Long Beach Long Beach, California Josefina V. Tinajero, Ph.D. Dean, College of Education The University of Texas at El Paso El Paso, Texas
Graphing Calculator Ruth M. Casey Mathematics Teacher Department Chair Anderson County High School Lawrenceburg, Kentucky Jerry Cummins Former President National Council of Supervisors of Mathematics Western Springs, Illinois
Learning Disabilities Kate Garnett, Ph.D. Chairperson, Coordinator Learning Disabilities School of Education Department of Special Education Hunter College, CUNY New York, New York
Pre-AP Dixie Ross AP Calculus Teacher Pflugerville High School Pflugerville, Texas
Reading and Vocabulary Douglas Fisher, Ph.D. Director of Professional Development and Professor City Heights Educational Collaborative San Diego State University San Diego, California Lynn T. Havens Director of Project CRISS Kalispell School District Kalispell, Montana
v
Teacher Reviewers
Each Teacher Reviewer reviewed at least two chapters of the Student Edition, giving feedback and suggestions for improving the effectiveness of the mathematics instruction. Chrissy Aldridge
Peter K. Christensen
Matt Gowdy
Teacher Charlotte Latin School Charlotte, North Carolina
Mathematics/AP Teacher Central High School Macon, Georgia
Mathematics Teacher Grimsley High School Greensboro, North Carolina
Harriette Neely Baker
Rebecca Claiborne
Wendy Hancuff
Mathematics Teacher South Mecklenburg High School Charlotte, North Carolina
Mathematics Department Chairperson George Washington Carver High School Columbus, Georgia
Teacher Jack Britt High School Fayetteville, North Carolina
Danny L. Barnes, NBCT
Laura Crook
Ernest A. Hoke Jr.
Mathematics Teacher Speight Middle School Stantonsburg, North Carolina
Mathematics Department Chair Middle Creek High School Apex, North Carolina
Mathematics Teacher E.B. Aycock Middle School Greenville, North Carolina
Aimee Barrette
Dayl F. Cutts
Carol B. Huss
Special Education Teacher Sedgefield Middle School Charlotte, North Carolina
Teacher Northwest Guilford High School Greensboro, North Carolina
Mathematics Teacher Independence High School Charlotte, North Carolina
Karen J. Blackert
Angela S. Davis
Deborah Ivy
Mathematics Teacher Myers Park High School Charlotte, North Carolina
Mathematics Teacher Bishop Spaugh Community Academy Charlotte, North Carolina
Mathematics Teacher Marie G. Davis Middle School Charlotte, North Carolina
Patricia R. Blackwell
Sheri Dunn-Ulm
Lynda B. (Lucy) Kay
Mathematics Department Chair East Mecklenburg High School Charlotte, North Carolina
Teacher Bainbridge High School Bainbridge, Georgia
Mathematics Department Chair Martin Middle School Raleigh, North Carolina
Rebecca B. Caison
Susan M. Fritsch
Julia Kolb
Mathematics Teacher Walter M. Williams High School Burlington, North Carolina
Mathematics Teacher, NBCT David W. Butler High School Matthews, North Carolina
Mathematics Teacher/Department Chair Leesville Road High School Raleigh, North Carolina
Myra Cannon
Dr. Jesse R. Gassaway
M. Kathleen Kroh
Mathematics Department Chair East Davidson High School Thomasville, North Carolina
Teacher Northwest Guilford Middle School Greensboro, North Carolina
Mathematics Teacher Z.B. Vance High School Charlotte, North Carolina
vi
Tosha S. Lamar
Susan M. Peeples
Mathematics Instructor Phoenix High School Lawrenceville, Georgia
Retired 8th Grade Mathematics Teacher Richland School District Two Columbia, South Carolina
Mathematics Teacher Clarke Central High School Athens, Georgia
Kay S. Laster
Carolyn G. Randolph
Elizabeth Webb
8th Grade Pre-Algebra/Algebra Teacher Rockingham County Middle School Reidsville, North Carolina
Mathematics Department Chair Kendrick High School Columbus, Georgia
Mathematics Department Chair Myers Park High School Charlotte, North Carolina
Joyce M. Lee
Tracey Shaw
Jack Whittemore
Lead Mathematics Teacher National Teachers Teaching with Technology Instructor George Washington Carver High School Columbus, Georgia
Mathematics Teacher Chatham Central High School Bear Creek, North Carolina
C & I Resource Teacher Charlotte-Mecklenburg Schools Charlotte, North Carolina
Marjorie Smith
Angela Whittington
Mathematics Teacher Eastern Randolph High School Ramseur, North Carolina
Mathematics Teacher North Forsyth High School Winston-Salem, North Carolina
McCoy Smith, III
Kentucky Consultants
Mathematics Department Chair Sedgefield Middle School Charlotte, North Carolina
Amy Adams Cash
Susan Marshall Mathematics Chairperson Kernodle Middle School Greensboro, North Carolina
Alice D. McLean Mathematics Coach West Charlotte High School Charlotte, North Carolina
Portia Mouton Mathematics Teacher Westside High School Macon, Georgia
Elaine Pappas Mathematics Department Chair Cedar Shoals High School Athens, Georgia
Bridget Sullivan 8th Grade Mathematics Teacher Northeast Middle School Charlotte, North Carolina
Marilyn R. Thompson Geometry/Mathematics Vertical Team Consultant Charlotte-Mecklenburg Schools Charlotte, North Carolina
Gwen Turner
Mathematics Educator/Department Chair Bowling Green High School Bowling Green, Kentucky
Susan Hack, NBCT Mathematics Teacher Oldham County High School Buckner, Kentucky
Kimberly L. Henderson Hockney Mathematics Educator Larry A. Ryle High School Union, Kentucky
vii
Unit 1
The Tools of Algebra 1-1 Using a Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Reading Math: Translating Expressions into Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1-2 Numbers and Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1-3 Variables and Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Extend 1-3
Spreadsheet Lab: Expressions and Spreadsheets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Student Toolbox Prerequisite Skills • Get Ready for Chapter 1 25 • Get Ready for the Next Lesson 30, 36, 41, 47, 53, 59
Reading and Writing Mathematics • • • • •
Reading in the Content Area 37 Reading Math 31 Reading Math Tips 32, 33, 49 Vocabulary Link 37, 43, 44 Writing in Math 30, 36, 41, 47, 53, 58, 65
1-4 Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1-5 Variables and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1-6 Ordered Pairs and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Explore 1-7 Algebra Lab: Scatter Plots. . . . . . . . . . . . . . . . . . . . . . . . 60
1-7 Scatter Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Extend 1-7
Graphing Calculator Lab: Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Standardized Test Practice • Griddable 36 • Multiple Choice 30, 36, 41, 47, 51, 53, 59, 66 • Worked Out Example 50
H.O.T. Problems Higher Order Thinking • • • •
Challenge 30, 36, 41, 47, 53, 58, 65 Find the Error 35, 47 Number Sense 65 Open Ended 30, 35, 41, 47, 53, 58, 65 • Reasoning 36 • Select a Technique 53 • Which One Doesn’t Belong? 41
viii Kim Taylor/DK Limited/CORBIS
Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
Integers
2-1 Integers and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78 Explore 2-2 Algebra Lab: Adding Integers . . . . . . . . . . . . . . . . . . . . 84
2-2 Adding Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Reading Math: Learning Mathematics Vocabulary . . . . . . . . . . 91 Explore 2-3 Algebra Lab: Subtracting
Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2-3 Subtracting Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Explore 2-4 Algebra Lab: Multiplying Integers . . . . . . . . . . . . . . . . 99
2-4 Multiplying Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Explore 2-5 Algebra Lab: Dividing Integers . . . . . . . . . . . . . . . . . . 105
2-5 Dividing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 ASSESSMENT
Prerequisite Skills
Table of Contents
2-6 The Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Student Toolbox • Get Ready for Chapter 2 77 • Get Ready for the Next Lesson 83, 90, 97, 104, 110
Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Reading and Writing Mathematics
Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
• Reading in the Content Area 86 • Reading Math 91 • Reading Math Tips 78, 86, 100, 106, 113, 114 • Writing in Math 83, 90, 97, 104, 110, 115
Standardized Test Practice • Griddable 83, 104 • Multiple Choice 83, 90, 97, 102, 104, 110, 115 • Worked Out Example 101
H.O.T. Problems Higher Order Thinking • • • • • • •
Challenge 83, 90, 96, 110, 115 Find the Error 96, 115 Number Sense 82, 115 Open Ended 82, 90, 96, 110, 115 Select a Tool 104 Select a Technique 97 Which One Doesn’t Belong? 82
ix Kim Taylor/DK Limited/CORBIS
3
Equations 3-1 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3-2 Simplifying Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . 129 Explore 3-3 Algebra Lab: Solving Equations
Using Algebra Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3-3 Solving Equations by Adding or Subtracting . . . . . . . . . . . . . . . 136 3-4 Solving Equations by Multiplying or Dividing . . . . . . . . . . . . . . 141 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3-5 Solving Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Reading Math: Translating Verbal Problems into Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3-6 Writing Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Student Toolbox
3-7 Sequences and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Prerequisite Skills
3-8 Using Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
• Get Ready for Chapter 3 123 • Get Ready for the Next Lesson 128, 133, 140, 145, 151, 157, 161
Reading and Writing Mathematics • • • • •
Reading in the Content Area 136 Reading Math 152 Reading Math Tips 143, 162 Vocabulary Link 124, 129 Writing in Math 128, 133, 140, 145, 150, 157, 161, 167
Standardized Test Practice • Griddable 140, 151 • Multiple Choice 128, 133, 139, 140, 145, 151, 157, 161, 167 • Worked Out Example 138
H.O.T. Problems Higher Order Thinking • Challenge 128, 133, 140, 145, 150, 156, 161, 166 • Find the Error 128, 132, 156 • Number Sense 145, 157 • Open Ended 128, 132, 140, 145, 150, 156, 161, 166 • Reasoning 167 • Select a Technique 140 • Which One Doesn’t Belong? 132
x John Cancalosi/Stock Boston
Extend 3-8
Spreadsheet Lab: Perimeter and Area . . . . . . . . . . . . 168
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Unit Factors and Fractions
4
4-1 Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Extend 4-1
Algebra Lab: Base 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4-2 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4-3 Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4-4 Simplifying Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Reading Math: Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Student Toolbox
Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Prerequisite Skills
4-5 Multiplying and Dividing Monomials . . . . . . . . . . . . . . . . . . . . . 203 Extend 4-5
Algebra Lab: A Half-Life Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
• Get Ready for Chapter 4 179 • Get Ready for the Next Lesson 184, 190, 195, 200, 207, 213
4-6 Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Reading and Writing Mathematics
4-7 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
• • • • •
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Reading in the Content Area 187 Reading Math 201 Reading Math Tips 180, 181, 205 Vocabulary Link 186 Writing in Math 184, 190, 195, 200, 207, 212, 218
Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Standardized Test Practice • Griddable 184, 218 • Multiple Choice 184, 190, 195, 198, 200, 207, 213, 218 • Worked Out Example 197–198
H.O.T. Problems Higher Order Thinking • Challenge 184, 189, 195, 200, 206, 212, 218 • Find the Error 189, 195 • Number Sense 189, 212, 218 • Open Ended 184, 189, 195, 200, 206, 212, 218 • Reasoning 207, 212 • Select a Tool 184 • Which One Doesn’t Belong? 200
xi John Cancalosi/Stock Boston
5
Rational Numbers 5-1 Writing Fractions as Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5-2 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5-3 Multiplying Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5-4 Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5-5 Adding and Subtracting Like Fractions . . . . . . . . . . . . . . . . . . . . 250 Reading Math: Factors and Multiples . . . . . . . . . . . . . . . . . . . 255 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5-6 Least Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Extend 5-6
Algebra Lab: Juniper Green . . . . . . . . . . . . . . . . . . . . 262
5-7 Adding and Subtracting Unlike Fractions . . . . . . . . . . . . . . . . . . 263
Student Toolbox Prerequisite Skills • Get Ready for Chapter 5 227 • Get Ready for the Next Lesson 233, 237, 244, 249, 254, 261, 272
5-8 Solving Equations with Rational Numbers . . . . . . . . . . . . . . . . . 268 Explore 5-9 Algebra Lab: Analyzing Data . . . . . . . . . . . . . . . . . . . 273
5-9 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Extend 5-9
Graphing Calculator Lab:
Mean and Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Reading and Writing Mathematics • • • • •
Reading in the Content Area 236 Reading Math 255 Reading Math Tips 234, 235, 245 Vocabulary Link 228 Writing in Math 233, 237, 244, 249, 254, 261, 267, 272, 278
Standardized Test Practice • Griddable 249, 272 • Multiple Choice 233, 237, 244, 249, 254, 261, 272, 277 • Worked Out Example 276–277
H.O.T. Problems Higher Order Thinking • Challenge 233, 237, 244, 249, 253, 261, 266, 272, 278 • Find the Error 243, 253, 267, 271 • Number Sense 233 • Open Ended 232, 237, 243, 249, 253, 261, 266, 278 • Reasoning 237 • Select a Technique 233 • Which One Doesn’t Belong? 272
xii Craig Tuttle/CORBIS
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Unit 3 6
Ratio, Proportion, and Percent
6-1 Ratios and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6-2 Proportional and Nonproportional Relationships . . . . . . . . . . . 297 Reading Math: Making Comparisons . . . . . . . . . . . . . . . . . . . 301 6-3 Using Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Extend 6-3
Algebra Lab: Capture-Recapture . . . . . . . . . . . . . . . . 307
6-4 Scale Drawings and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 6-5 Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . . . . . . . . 313 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Explore 6-6 Algebra Lab: Using a Percent Model . . . . . . . . . . . . . 320
6-6 Using the Percent Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Student Toolbox Prerequisite Skills • Get Ready for Chapter 6 291 • Get Ready for the Next Lesson 296, 300, 306, 312, 318, 326, 331, 336, 342
6-7 Finding Percents Mentally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6-8 Using Percent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Extend 6-8
Spreadsheet Lab: Compound Interest . . . . . . . . . . . . 337
6-9 Percent of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 6-10 Using Sampling to Predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Reading and Writing Mathematics • • • •
Reading in the Content Area 297 Reading Math 301, 313 Reading Math Tips 298, 303, 334 Writing in Math 296, 300, 306, 312, 317, 326, 331, 336, 342, 347
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
Standardized Test Practice • Griddable 326 • Multiple Choice 296, 300, 306, 312, 318, 326, 331, 336, 340, 342, 347 • Worked Out Example 339–340
H.O.T. Problems Higher Order Thinking • Challenge 300, 306, 312, 317, 326, 331, 336, 341, 347 • Find the Error 312, 341 • Number Sense 317 • Open Ended 296, 300, 306, 312, 317, 326, 331, 336, 341, 347 • Reasoning 331 • Select a Technique 296 • Which One Doesn’t Belong? 317, 341
xiii Craig Tuttle/CORBIS
7
Functions and Graphing Explore 7-1 Algebra Lab: Input and Output . . . . . . . . . . . . . . . . . 358
7-1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Extend 7-1
Graphing Calculator Lab: Function Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7-2 Representing Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Reading Math: Language of Functions . . . . . . . . . . . . . . . . . . . 370 7-3 Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7-4 Constant Rate of Change and Direct Variation . . . . . . . . . . . . . . 376 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Explore 7-5 Algebra Lab: It’s All Downhill . . . . . . . . . . . . . . . . . . . 383
7-5 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Student Toolbox Prerequisite Skills • Get Ready for Chapter 7 357 • Get Ready for the Next Lesson 363, 366, 375, 381, 389, 394, 402
Reading and Writing Mathematics • • • • •
Reading in the Content Area 359 Reading Math 370 Reading Math Tips 365, 366, 391 Vocabulary Link 360 Writing in Math 363, 369, 375, 381, 389, 394, 402, 406
Standardized Test Practice • Multiple Choice 363, 369, 375, 381, 387, 389, 394, 402, 406 • Worked Out Example 386
H.O.T. Problems Higher Order Thinking • Challenge 363, 369, 375, 388, 394, 402, 406 • Find the Error 388, 394 • Number Sense 369 • Open Ended 362, 369, 375, 381, 388, 394, 401, 406 • Reasoning 362, 381, 389 • Select a Tool 401
xiv Bill Ross/CORBIS
Extend 7-5
Graphing Calculator Lab: Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
7-6 Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Extend 7-6
Graphing Calculator Lab: The Family of Linear Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
7-7 Writing Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 7-8 Prediction Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
8
Equations and Inequalities
Explore 8-1 Algebra Lab: Equations with
Variables on Each Side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 8-1 Solving Equations with Variables on Each Side . . . . . . . . . . . . . 420 8-2 Solving Equations with Grouping Symbols . . . . . . . . . . . . . . . . . 424 Reading Math: Meanings of At Most and At Least . . . . . . . . . 429 8-3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 8-4 Solving Inequalities by Adding or Subtracting . . . . . . . . . . . . . . 435 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 8-5 Solving Inequalities by Multiplying or Dividing . . . . . . . . . . . . . 441 8-6 Solving Multi-Step Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . 446 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
Student Toolbox Prerequisite Skills • Get Ready for Chapter 8 417 • Get Ready for the Next Lesson 423, 428, 434, 439, 445
Reading and Writing Mathematics • • • •
Reading in the Content Area 430 Reading Math 429 Reading Math Tips 431 Writing in Math 423, 428, 434, 439, 445, 449
Standardized Test Practice • Griddable 445 • Multiple Choice 423, 428, 434, 439, 444, 445, 450 • Worked Out Example 442
H.O.T. Problems Higher Order Thinking • Challenge 423, 427, 434, 439, 444, 449 • Find the Error 439, 445, 449 • Number Sense 422, 428, 434 • Open Ended 423, 427, 434, 439, 444, 449 • Select a Tool 428 • Select a Technique 428
xv Bill Ross/CORBIS
Unit 4 9
Real Numbers and Right Triangles Explore 9-1 Algebra Lab: Squares and Square Roots . . . . . . . . . . 462
9-1 Squares and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 9-2 The Real Number System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Reading Math: Learning Geometry Vocabulary. . . . . . . . . . . . 475
Student Toolbox Prerequisite Skills • Get Ready for Chapter 9 461 • Get Ready for the Next Lesson 468, 474, 481, 490, 496
Reading and Writing Mathematics • Reading in the Content Area 464 • Reading Math 475, 497 • Reading Math Tips 464, 465, 476, 478, 485, 493 • Writing in Math 468, 474, 480, 490, 496, 502
9-3 Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Explore 9-4 Algebra Lab: The Pythagorean
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 9-4 The Pythagorean Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Extend 9-4
Algebra Lab: Graphing Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
9-5 The Distance Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 9-6 Similar Figures and Indirect Measurement . . . . . . . . . . . . . . . . 497 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Standardized Test Practice
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
• Griddable 488, 490 • Multiple Choice 468, 474, 481, 490, 496, 502 • Worked Out Example 486–487
Standardized Test Practice. . .
H.O.T. Problems Higher Order Thinking • Challenge 468, 473, 480, 490, 495, 502 • Find the Error 490, 502 • Number Sense 468 • Open Ended 468, 473, 480, 490, 495, 502 • Reasoning 468 • Select a Tool 480 • Select a Technique 495 • Which One Doesn’t Belong? 474
xvi CORBIS
. . . . . . . . . . . . . . . . . . . 508
10 Two-Dimensional Figures 10-1 Line and Angle Relationships. . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 10-2 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 10-3 Transformations on the Coordinate Plane . . . . . . . . . . . . . . . . . 524 Extend 10-3 Geometry Lab: Rotations . . . . . . . . . . . . . . . . . . . . . . 531
10-4 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Reading Math: Learning Mathematics Prefixes . . . . . . . . . . . . 538 10-5 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Extend 10-5 Geometry Lab: Tessellations. . . . . . . . . . . . . . . . . . . . 544
10-6 Area: Parallelograms, Triangles, and Trapezoids . . . . . . . . . . . 545 10-7 Circles: Circumference and Area . . . . . . . . . . . . . . . . . . . . . . . . 551 Extend 10-7 Spreadsheet Lab: Circle Graphs
and Spreadsheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 10-8 Area: Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 Extend 10-8 Spreadsheet Lab: Dilations and
Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
Student Toolbox Prerequisite Skills • Get Ready for Chapter 10 511 • Get Ready for the Next Lesson 517, 523, 530, 536, 543, 550, 556
Reading and Writing Mathematics • • • • •
Reading in the Content Area 512 Reading Math 538 Reading Math Tips 513, 515, 525 Vocabulary Link 518 Writing in Math 517, 523, 529, 536, 543, 550, 556, 562
Standardized Test Practice • Griddable 536, 562 • Multiple Choice 517, 523, 527, 530, 536, 543, 550, 556, 562 • Worked Out Example 525
H.O.T. Problems Higher Order Thinking • Challenge 517, 523, 529, 535, 543, 550, 556, 562 • Find the Error 523, 556 • Number Sense 556 • Open Ended 517, 529, 536, 543, 550, 556, 562 • Select a Tool 543 • Which One Doesn’t Belong? 529
xvii CORBIS
11 Three-Dimensional Figures Explore 11-1 Geometry Lab: Building Three-
Dimensional Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 11-1 Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Explore 11-2 Geometry Lab: Volume . . . . . . . . . . . . . . . . . . . . . . . 582
11-2 Volume: Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11-3 Volume: Pyramids, Cones, and Spheres . . . . . . . . . . . . . . . . . . . 589 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Explore 11-4 Geometry Lab: Exploring Lateral
Area and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 11-4 Surface Area: Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . 597
Student Toolbox Prerequisite Skills • Get Ready for Chapter 11 573 • Get Ready for the Next Lesson 581, 588, 594, 601, 606
11-5 Surface Area: Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . 602 Explore 11-6 Geometry Lab: Similar Solids . . . . . . . . . . . . . . . . . . . 607
11-6 Similar Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Reading Math: Precision and Accuracy . . . . . . . . . . . . . . . . . . . 614 ASSESSMENT
Reading and Writing Mathematics
Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
• • • •
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
Reading in the Content Area 583 Reading Math 614 Vocabulary Link 576, 597 Writing in Math 580, 588, 594, 601, 606, 613
Standardized Test Practice • Griddable 588 • Multiple Choice 581, 586, 588, 594, 601, 606, 613 • Worked Out Example 584–585
H.O.T. Problems Higher Order Thinking • Challenge 580, 588, 594, 601, 606, 613 • Find the Error 587, 612 • Number Sense 612 • Open Ended 580, 587, 593, 601, 605, 612 • Reasoning 580 • Select a Technique 593
xviii Antonio M. Rosario/Getty Images
Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
Unit 5 Statistics and 12 More Probability 12-1 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Extend 12-1 Graphing Calculator Lab: Stem-and-
Leaf Plots and Line Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 12-2 Measures of Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 12-3 Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Extend 12-3 Graphing Calculator Lab: Box-and-
Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 12-4 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Extend 12-4 Graphing Calculator Lab: Histograms. . . . . . . . . . . . 650
12-5 Selecting an Appropriate Display . . . . . . . . . . . . . . . . . . . . . . . . 651 Extend 12-5 Spreadsheet Lab: Bar Graphs and
Student Toolbox Prerequisite Skills • Get Ready for Chapter 12 625 • Get Ready for the Next Lesson 631, 637, 642, 648, 656, 663, 669, 674, 680
Line Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 12-6 Misleading Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 Reading Math: Dealing with Bias . . . . . . . . . . . . . . . . . . . . . . . 664 12-7 Simple Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 12-8 Counting Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 Extend 12-8 Algebra Lab: Probability and
Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 12-9 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 676 Explore 12-10 Graphing Calculator Lab: Probability
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 12-10 Probability of Composite Events. . . . . . . . . . . . . . . . . . . . . . . . . 682 Extend 12-10 Algebra Lab: Simulations . . . . . . . . . . . . . . . . . . . . . . 688
ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
Reading and Writing Mathematics • Reading in the Content Area 633 • Reading Math 664 • Reading Math Tips 634, 639, 645, 659, 660, 666, 676, 678, 682 • Writing in Math 631, 637, 642, 648, 655, 663, 669, 674, 680, 686
Standardized Test Practice • Griddable 642 • Multiple Choice 631, 637, 642, 648, 654, 656, 663, 669, 674, 680, 687 • Worked Out Example 653
H.O.T. Problems Higher Order Thinking • Challenge 630, 637, 642, 648, 655, 663, 669, 674, 680, 686 • Find the Error 630, 679, 686 • Number Sense 642 • Open Ended 636, 655, 663, 669, 674, 680, 686 • Reasoning 648, 674 • Select a Tool 655
xix
13 Polynomials and Nonlinear Functions Reading Math: Prefixes and Polynomials . . . . . . . . . . . . . . . . . 700 13-1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Extend 13-1 Algebra Lab: Modeling Polynomials
with Algebra Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 13-2 Adding Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 13-3 Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Explore 13-4 Algebra Lab: Modeling
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 13-4 Multiplying a Polynomial by a Monomial . . . . . . . . . . . . . . . . . . 716 13-5 Linear and Nonlinear Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 720 13-6 Graphing Quadratic and Cubic Equations. . . . . . . . . . . . . . . . . . 726
Student Toolbox Prerequisite Skills • Get Ready for Chapter 13 699 • Get Ready for the Next Lesson 704, 709, 713, 719, 725
Reading and Writing Mathematics • • • •
Reading in the Content Area 701 Reading Math 700 Reading Math Tips 722 Writing in Math 704, 709, 713, 719, 724, 729
Standardized Test Practice • Multiple Choice 704, 709, 713, 718, 719, 725, 730 • Worked Out Example 717
H.O.T. Problems Higher Order Thinking • Challenge 704, 709, 713, 719, 724, 729 • Find the Error 704, 709 • Number Sense 729 • Open Ended 709, 713, 719, 724, 729 • Reasoning 719, 729 • Select a Tool 713 • Which One Doesn’t Belong? 724
xx Age fotostock/SuperStock
Extend 13-6 Graphing Calculator Lab: The Family of
Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Standardized Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736
Student Handbook Built-In Workbooks Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761 Mixed Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 Extend 10-6: Graphing Geometric Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 807 Preparing for Standardized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Reference English-Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R1 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R24 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R58 Photo Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R77
xxi Age fotostock/SuperStock
Start Smart
tart mart Be a Better Problem Solver As you gear up to study mathematics, you are probably wondering, “What will I learn this year?” There are three focal points this year:
•
Use basic principles of algebra to analyze and represent proportional and nonproportional linear relationships,
• •
Apply operations with rational numbers, and Use probability and statistics to make predictions.
Along the way, you’ll learn more about problem solving, how to use the tools and language of mathematics, and how to THINK mathematically.
Start Smart 1 Digital Vision/Getty Images
Problem-Solving Strategy: Look for a Pattern R e a l - Wo r l d Problem Solving Tuning musical instruments is a science that also involves mathematics. All music is caused by vibrations. The number of vibrations per second is called the frequency. There is a pattern in the frequencies: the frequency of any note is 1.059 times the frequency of the note that is one-half step lower. There are many problem-solving strategies in mathematics. One of the most common is to look for a pattern. To use this strategy, analyze the first few numbers in a pattern and identify a rule that is used to go from the first number in the pattern to the second, and then to the third, and so on. Then use the rule to extend the pattern and find a solution.
E-MAIL Ramon got an E-mail from his friend Angela. After 10 minutes, he forwarded it to 2 of his friends. After 10 more minutes, those 2 friends forwarded it to 2 more friends. If the message was forwarded like this every 10 minutes, how many people received Angela’s E-mail message after 40 minutes?
1
EXPLORE What are you trying to find? Restate the problem in your own words. Use as few words as possible. You need to find the total number of people who received the E-mail. What other information do you need? Do you think you’ll need any additional information such as a formula or a measurement conversion? You do not need any additional information.
2 Start Smart David Young-Wolff/PhotoEdit
3
PLAN
SOLVE
Organize the data in a table. Look for a pattern in the data. Then extend the pattern. 1,
2, ×2
,
4, ×2
×2
, ×2
To continue the pattern, multiply each term by 2. 4×2=8
Time (min)
People Receiving Message
1
10
2
20
4
30
40
Start Smart
2
8 × 2 = 16
So, 1 + 2 + 4 + 8 + 16 or 31 people got the message.
4
CHECK
In the 40th minute, 16 people received the E-mail. Half as many received it each time before that. So, it is reasonable that the total will be less than 16 × 2 or 32.
Practice Solve each problem by looking for a pattern. 1. List the first five common multiples of 3, 4, and 6. Write an expression to
describe all common multiples of 3, 4, and 6. 2. Two workers can make two chairs in two days. How many chairs can
8 workers working at the same rate make in 20 days? 3. Courtney travels south on her bicycle riding
8 miles per hour. One hour later, her friend Horacio starts riding his bicycle from the same location. If he travels south at 10 miles per hour, how long will it take him to catch Courtney? 4. What is the perimeter of the twelfth figure?
Figure 1 Perimeter ⫽ 6
Figure 2 Perimeter ⫽8
Figure 3 Perimeter ⫽10
5. A ball bounces back 0.6 of its height on
every bounce. If a ball is dropped from 200 feet, how high does it bounce on the fifth bounce? Round to the nearest tenth. 6. A forest fire spread to 41 acres in 10 hours.
Each hour the fire spread to four more acres than the previous hour. How many acres were consumed during each hour of the fire?
Problem-Solving Strategy: Look for a Pattern John Evans
3
Problem-Solving Strategy: Make a Table or List One strategy for solving problems is to make a table or list. A table allows you to organize information in an understandable way. When you make a list, use an organized approach so you do not leave out important items.
A fruit machine accepts dollars, and each piece of fruit costs 65 cents. If the machine gives only nickels, dimes, and quarters, what combinations of those coins are possible as change for a dollar? The machine will give back $1.00 - $0.65 or 35 cents in change in a combination of nickels, dimes, and quarters. Make a table showing different combinations of nickels, dimes, and quarters that total 35 cents. Organize the table by starting with the combinations that include the most quarters. quarters
dimes
nickels
1
1
1
2
3
1
2
3
1
5
7
The total for each combination of the coins is 35 cents. There are 6 combinations possible.
Practice Solve each problem by making a table or list. 1. How many ways can you make change for a half-dollar using only nickels,
dimes, and quarters? 2. A number cube has faces numbered 1 to 6. If a red and a blue cube are
tossed and the faces landing up are added, how many ways can you roll a sum less than 8? 3. A penny, a nickel, a dime, and a quarter are in a purse. How many
amounts of money are possible if you grab two coins at random? 4. The three counters at the right are used for
a board game. If the counters are tossed, how many ways can at least one counter with Side A turn up? 5. How many ways can you receive change for a
Counters
Side 1 Side 2
Counter 1
A
B
Counter 2
A
C
Counter 3
B
C
quarter if at least one coin is a dime? 6. Jorge had 55 football cards. He traded 8 cards for 5 from Elise. He traded 6
more for 4 from Leon and 5 for 3 from Bret. Finally, he traded 12 cards for 9 from Ginger. How many cards does Jorge have now? 4 Start Smart
Start Smart
Problem-Solving Strategy: Work Backward In most problems, a set of conditions or facts is given and an end result must be found. However, some problems start with the result and ask for something that happened earlier. The strategy of working backward can be used to solve problems like this. To use this strategy, start with the end result and undo each step.
Kendrick spent half of the money he had this morning on lunch. After lunch, he loaned his friend a dollar. Now he has $1.50. How much money did Kendrick start with? Start with the end result, $1.50, and work backward to find the amount Kendrick started with. Kendrick now has $1.50. $1.50 Add $1.00 to undo Undo the $1 he loaned to his friend. +1.00 giving his friend $1.00. $2.50 Multiply by 2 to Undo the half he spent for lunch. × 2 undo spending half $5.00 the original amount. The amount Kendrick started with was $5.00. CHECK
Kendrick started with $5.00. If he spent half of that, or $2.50, on lunch and loaned his friend $1.00, he would have $1.50 left. This matches the amount stated in the problem, so the solution is correct.
Practice Solve each problem by working backward. 1. Tia used half of her allowance to buy a ticket to the class play. Then she
spent $1.75 for an ice cream cone. Now she has $2.25 left. How much is her allowance? 2. Lawanda put $15 of her paycheck in savings. Then she spent one-half of
what was left on clothes. She paid $24 for a concert ticket and later spent one-half of what was then left on a book. When she got home, she had $14 left. What was the amount of Lawanda’s paycheck? 3. A certain number is multiplied by 3, and then 5 is added to the result. The
final answer is 41. What is the number? 4. Mr. and Mrs. Delgado each own an equal number of shares of a stock.
Mr. Delgado sells one-third of his shares for $2700. What was the total value of Mr. and Mrs. Delgado’s stock before the sale? 5. A certain bacteria doubles its population every 12 hours. After 3 full days,
there are 1600 bacteria in a culture. How many bacteria were there at the beginning of the first day? 6. To catch a 7:30 A.M. bus, Don needs 30 minutes to get dressed, 30 minutes
for breakfast, and 15 minutes to walk to the bus stop. What time should he wake up? Problem-Solving Strategy: Work Backward PhotoLink/Getty Images
5
Problem-Solving Strategy: Guess and Check To solve some problems, you can make a reasonable guess and then check it in the problem. You can then use the results to improve your guess until you find the solution. This strategy is called guess and check.
The product of two consecutive even integers is 1088. What are the integers? The product is close to 1000. Make a guess. Let’s try 24 and 26.
24 × 26 = 624
This product is too low.
Adjust the guess upward. Try 30 and 32.
30 × 32 = 960
This product is still too low.
Adjust the guess upward again. Try 34 and 36.
34 × 36 = 1224
This product is too high.
Try between 30 and 34. Try 32 and 34.
32 × 34 = 1088
This is the correct product.
The integers are 32 and 34.
Practice Use the guess-and-check strategy to solve each problem. 1. The product of two consecutive odd integers is 783. What are the integers? 2. Brianne is three times as old as Camila. Four years from now she will be
just two times as old as Camila. How old are Brianne and Camila now? 3. Rafael is burning a CD for Selma. The CD will hold 35 minutes of music.
Which songs should he select from the list to record the maximum time on the CD without going over? Song
A
B
C
D
E
F
G
H
I
J
Time
5 min 4s
9 min 10 s
4 min 12 s
3 min 9s
3 min 44 s
4 min 30 s
5 min 0s
7 min 21 s
4 min 33 s
5 min 58 s
4. Each hand in the human body has 27 bones. There are 6 more bones in the
fingers than in the wrist. There are 3 fewer bones in the palm than in the wrist. How many bones are in each part of the hand? 5. The Science Club sold candy bars and soft pretzels to raise money for an
animal shelter. They raised a total of $62.75. They made 25¢ profit on each candy bar and 30¢ profit on each pretzel sold. How many of each did they sell? 6. Odell has the same number of quarters, dimes, and nickels. In all he has $4
in change. How many of each coin does he have? 7. Anita sold tickets to the school musical. She had 12 bills worth $175 for the
tickets she sold. If all the money was in $5 bills, $10 bills, and $20 bills, how many of each bill did she have? 6 Start Smart
Start Smart
Problem-Solving Strategy: Solve a Simpler Problem One of the strategies you can use to solve a problem is to solve a simpler problem. To use this strategy, first solve a simpler or more familiar case of the problem. Then use the same concepts and relationships to solve the original problem.
Find the sum of the numbers 1 through 500. Consider a simpler problem. Find the sum of the numbers 1 through 10. Notice that you can group the addends into partial sums as shown below. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 11 The number of sums is 5, or half the number of addends. 11 11 11 Each partial sum is 11, the sum 11 of the first and last numbers. The sum is 5 × 11 or 55. Use the same concepts to find the sum of the numbers 1 through 500. 1 + 2 + 3 + … + 499 + 500 = 250 × 501 Multiply half the number of addends, 250, by the sum of = 125,250 the first and last numbers, 501.
Practice Solve each problem by first solving a simpler problem. 1. Find the sum of the numbers 1 through 1000. 2. Find the number of squares of any size in the
game board shown at the right. 3. How many links are needed to join 30 pieces of
chain into one long chain? 4. Three people can pick six baskets of apples in one
hour. How many baskets of apples can 12 people pick in one-half hour? 5. A shirt shop has 112 orders for T-shirt designs.
Three designers can make 12 shirts in 2 hours. How many designers are needed to complete the orders in 8 hours? 6. Stamps for postcards cost $0.24, and stamps for first-class letters cost $0.39.
Diego wants to send postcards and letters to 10 friends. If he has $3.15 for stamps, how many postcards and how many letters can he send? Problem-Solving Strategy: Solve a Simpler Problem
7
Problem-Solving Strategy: Draw a Diagram Another strategy for solving problems is to draw a diagram. There will be times when a sketch or diagram will give you a better picture of how to tackle a mathematics problem. Adding details like units, labels, and numbers to the drawing or sketch can help you make decisions on how to solve the problem.
Imani is trying to determine the number of 9-inch tiles needed to cover her patio. The rectangular patio measures 8-feet by 10-feet. What is the minimum number of 9-inch tiles Imani should purchase? First, draw a diagram of the situation. Express the measurement of the patio in inches. If each tile is 9 inches square, the minimum number of tiles for the width of the patio is 96 ÷ 9 ≈ 10.7 or 11 tiles. The minimum number of tiles for the length of the patio is 120 ÷ 9 ≈ 13.3 or 14 tiles.
0ATIO
FT IN
So, the minimum number of tiles Imani needs to cover the patio is 11 × 14 or 154 tiles.
Practice Solve each problem by drawing a diagram. 1. The area of a rectangular flower bed is 24 square feet. If the sides are
whole number dimensions, how many combinations of lengths and widths are possible for the flower bed? List them. 2. Kevin was hired to paint a mural on a wall that measures 15 feet by
20 feet. Starting from the center of the wall, he will paint a square that measures 3 feet. The dimensions of the next square will be 1.5 times greater than and centered on the previous square. How many squares can Kevin paint on the wall? 3. A 500-gallon bathtub is being filled with water. Eighty gallons of water are
in the bathtub after 4 minutes. How long will it take to fill the bathtub? 4. It takes 42 minutes to cut a 2-inch by 4-inch piece of wood into 7 equally
sized pieces. How long will it take to cut a similar 2-inch by 4-inch into 4 equally sized pieces? 5. Find the number of line segments that can be drawn between any two
vertices of an octagon. 6. How many different teams of 3 players can be chosen from 8 players?
8 Start Smart
FT IN
Start Smart
Problem-Solving Strategy: Act It Out There are problem situations where acting the problem out will provide you with a visual image and provide direction for solving the problem. To act it out, you could use real people, models of the real objects, or manipulatives like tiles and cubes, which will help you visualize the problem and make decisions about how to proceed with solving the problem.
A quiz has 5 multiple-choice items each with choices A, B, C, and D. Is spinning a spinner with 4 equal sections to decide the answers a good strategy for taking the quiz? First, make a spinner with 4 equal sections and use it to act out taking the quiz. Each section represents one of the choices A, B, C, and D for each question. Suppose the correct answers to the quiz are D, D, B, C, and A. Do 5 trials. Answers
D
D
B
C
A
Number Correct
Trial 1
D
A
D
B
C
1
Trial 2
C
B
C
A
B
Trial 3
B
C
D
A
A
1
Trial 4
B
D
A
C
B
2
Trial 5
D
A
B
A
D
2
The experiment only produces 1 or 2 correct answers. So, spinning a spinner is not a good strategy to use to answer a multiple-choice quiz with 4 choices.
Practice Solve each problem by acting the problem out. 1. A test has 10 true-false questions. Is tossing a coin to decide the answers a
good strategy for taking the test? Explain. 2. A pizza parlor has thin crust and deep dish, 2 different cheeses, and
4 toppings. How many different one-cheese and one-topping pizzas can be ordered? 3. A field hockey conference has four teams. In how many ways can first,
second, and third place be awarded at the end of the season? 4. Waban makes 2 out of every 3 free throws he attempts. How many would
you expect him to make in his next 25 attempts? 5. A gumball machine has an equal number of red, yellow, and orange
gumballs. If it costs $0.25 per gumball, how much would you have to spend to get at least one gumball of each color? Problem-Solving Strategy: Act It Out
9
The Graphing Calculator This year, you may use an exciting tool to help you solve problems—a graphing calculator. Graphing Calculator Labs have been included in your textbook so you can use technology to explore concepts. These labs use the TI-83 Plus or TI-84 Plus calculator. A graphing calculator does more than just graph. You can also use it to calculate.
is used to enter equations.
Press 2nd to access the additional functions listed above each key.
The key is used to find the second power of a number or expression.
Press ON to turn on your calculator. Press 2nd [OFF] to turn off your calculator.
( ) is used to
indicate a negative or opposite value.
10 Start Smart Matt Meadows
Press 2nd [TABLE] to display a table of values for equations entered key. using the
Press CLEAR once to clear an entry. Press CLEAR twice to clear the screen. Use the operation keys to add, subtract, multiply or divide. Multiplication is shown as * on the screen and division is displayed as /.
The ENTER key acts like an equals button to evaluate an expression. It is also used to select menu items.
Start Smart
EXAMPLE
ntering and Evaluating Expressions
Evaluate each expression. a. -2[4(11 - 6)] + (-23) KEYSTROKES:
( ) 2
4
11
( )
6
ENTER
23
b. 3(-4) + [(12 ÷ 6) - 5(-8)] KEYSTROKES:
12 ⫼ 6
( ) 4
3
5
( ) 8
ENTER
Evaluate each expression. 1. 8 + [7(12 ÷ 4)]
2. 5[(5 + 14) - 2(7)]
3. [6(8 ÷ 12)] × 3
4. 5 + [(8 × 2) - 7]
5. 4[3(21 - 17) + 3]
6. 7[5 + (13 - 4) ÷ 3]
EXAMPLE
owers and Exponents
Evaluate each expression. a. 52 × 23 KEYSTROKES:
⫻ 2
5
3 ENTER
b. (-4)3 ÷ 25 3 ⫼ 2
( ) 4
KEYSTROKES:
5 ENTER
Evaluate each expression. 7. 38
8. 2 · 63
10. 54 - 33
EXAMPLE
9. 43 · 27
11. 3 · 25 · 45
12. 3 · 53 + 4 · 25
oots
Evaluate each expression. 625 + 5 a. - √ KEYSTROKES:
( )
[ √ ] 625
5 ENTER
b. √ 62 + 82 KEYSTROKES:
[ √ ] 6
8
ENTER
Evaluate each expression. 13. √ 144 + 26
14. √ 324 - 12
15. - √ 225 + 6
16. - √ 169 - 3
17. √ 32 + 7
18. √ 142 + 27
The Graphing Calculator
11
Choose the Best Method of Computation Solving problems is more than using paper and pencil. Follow the path to choose the best method of computation.
12 Start Smart (tl)Image100 Ltd, (tr)Monica Lau/Getty Images, (bl)RubberBall/PictureQuest, (br)Digital Vision/PunchStock
Start Smart
Practice Choose the best method of computation to solve each problem. Then solve. 1. Kimi paid $677.48, including tax, for a surfboard. She then decided to have
a protective spray applied to her board for $47.98. What was the total cost of the surfboard and the spray? 2. An average-sized orca whale eats about 551 pounds of food a day. How
many pounds of food will an orca whale eat in one year? For Exercises 3-5, use the information in the table that shows the sales of Café Mocha’s coffee of the month.
Coffee of the Month Sales
3. How many more cups of this kind of
coffee did Café Mocha sell in January than in April? 4. If Café Mocha charges $1.98 for each
Month
Number of Cups Sold
January
850
February
765
March
587
April
500
May
387
cup of coffee, about how much money did Café Mocha earn in March? 5. In the month of May, Café Mocha had to raise their prices for each cup
of coffee to $2.25. How much money did Café Mocha earn in the month of May? 6. The Student Council is making pizzas to sell at the football game on
1 Friday. Each pizza needs 2_ cups of cheese. If the student council members 4
make 25 pizzas, how many cups of cheese will they need? 7. The length of Fun Center’s go-kart track is 843 feet. If Nadia circled the
track 9 times, about how many feet did she travel? 8. Use the table below to find the total area of the Great Lakes. Great Lakes Great Lake
Area (mi2)
Lake Superior
31,698
Lake Huron
23,011
Lake Michigan
22,316
Lake Erie
9922
Lake Ontario
7320
Source: worldatlas.com
9. Music Megastore was having a sale on blank CDs and DVDs. Ivan bought
a package of 300 blank CDs for $39 including tax. What is the individual cost of each CD? 10. An art supply store sells 5 different sized canvases. The surface area of the
middle-size canvas is 3.5 times larger than the surface area of the extrasmall canvas. If the surface area of the extra-small canvas is 81 square inches, what is the surface area of the middle-sized canvas? Choose the Best Method of Computation
13
Why do I need my math book? Have you ever been in class and not understood all of what was presented? Or, you understood everything in class, but at home, got stuck on how to solve a couple of problems? Maybe you just wondered when you were ever going to use this stuff?
These next few pages are designed to help you understand everything your math book can be used for … besides homework problems! Before you read, have a goal. • What information are you trying to find? • Why is this information important to you? • How will you use the information? Have a plan when you read. • • • •
Read the Main Idea at the beginning of the lesson. Look over photos, tables, graphs, and opening activities. Locate words highlighted in yellow and read their definitions. Find Key Concept and Concept Summary boxes for a preview of what’s important. • Skim the example problems. Keep a positive attitude. • Expect mathematics reading to take time. • It is normal to not understand some concepts the first time. • If you don’t understand something you read, it is likely that others don’t understand it either.
14 Start Smart John Evans
Start Smart
Doing Your Homework Regardless of how well you paid attention in class, by the time you arrive at home, your notes may no longer make any sense and your homework seems impossible. It’s during these times that your book can be most useful. • Each lesson has example problems, solved step-by-step, so you can review the day’s lesson material. •
has extra examples at pre-alg.com to coach you through solving those difficult problems.
• Each exercise set has HOMEWORK HE H LPP boxes that show you which examples may help with your homework problems. • Answers to the odd-numbered problems are in the back of the book. Use them to see if you are solving the problems correctly. If you have difficulty on an even problem, do the odd problem next to it. That should give you a hint about how to proceed with the even problem.
Doing Your Homework John Evans
15
Studying for a Test You may think there is no way to study for a math test! However, there are ways to review before a test. Your book can help! • Review all of the new vocabulary words and be sure you understand their definitions. These can be found on the first page of each lesson or highlighted in yellow in the text. • Review the notes you’ve taken on your questions that you still need answered.
and write down any
• Practice all of the concepts presented in the chapter by using the chapter Study Guide and Review. It has additional problems for you to try as well as more examples to help you understand. You can also take the Chapter Practice Test. • Take the Self-check Quizzes at pre-alg.com.
16 Start Smart John Evans
Start Smart
Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1. What is the title of Chapter 1? 2. How can you tell what you’ll learn in Lesson 1-1? 3. What is the key concept presented in Lesson 1-2? 4. List the new vocabulary words that are presented in
Lesson 1-3. 5. In the margin of Lesson 1-3, there is a Vocabulary Link.
What can you learn from that feature? 6. How many examples are presented in Lesson 1-3? 7. What is the web address where you could find extra
examples? 8. Suppose you’re doing your homework on page 40 and you
get stuck on Exercise 19. Where could you find help? 9. What is the title of the feature in Lesson 1-5 that tells you
how to read the ≠ symbol? 10. What is the title of the feature in Lesson 1-6 that tells you
about the units on the x- and y-axes? 11. Sometimes you may ask “When am I ever going to use
this”? Name a situation that uses the concepts from Lesson 1-7. 12. There is a Real-World Career mentioned in Lesson 1-7.
What is it? 13. What is the web address that would allow you to take a
self-check quiz to be sure you understand the lesson? 14. On what pages will you find the Study Guide and Review
for Chapter 1? 15. Suppose you can’t figure out how to do Exercise 35 in the
Study Guide on page 71. Where could you find help?
Scavenger Hunt
17
The following pages contain data that you’ll use throughout the book. JEFFERSON CIT Y, MISSOU
RI
-*
i`>À ÌÞ 7AIN 3
TREET
-ISSOURI "LVD
University of Akron ege Columbus State Community Coll Ohio University Sinclair University University of Toledo Bowling Green State University
FF 2
OAD
,' 2E
X7
5NIO N
HIT
TON
0ACIF IC
2AILR
%X PY
OA D
#F
2E
D7
HA
LEY
%X PY
Universities 2004 Ten Largest Ohio Colleges and Enrollment College or University
Cuyahoga Community College Kent State University
"LU
*EFFERSON #ITY -EMORIAL !IRPORT
ivviÀÃ ÌÞ
HIGHER EDUCATION
Ohio State University University of Cincinnati
0+
50,995
,+
27,178 25,214 24,347 23,282 21,872 20,096 19,563 19,480 18,989
INDEPENDENCE SQUARE
Source: infoplease.com
Congress Hall
BOSTO N CREA M PIE
Recipe for:
Boston Cream 1(2 laye Pie r size) p kg yello 1 pkg in w cake mix stant va nilla pu 1 (1 oz ) dding m ix sq unsw eetened 1 tables chocolate poon bu tter 1 cup sif _1 ted pow dered su 2 teaspo gar on vanil la Source 3 – 4 : cooks.c teaspoo om ns wate r
18
United States Data File
Independence Hall
Philosophical Hall
Great Essentials exhibit West Wing Sansom Street
Old City Hall
Independence Hall tours start in East Wing
Library Hall
INDEPENDENCE 500 ft
SQUARE 375 ft
The Declaration of Independence was first publicly read at Independence Square in Philadelphia, Pennsylvania.
N
BOOMERANG
United States Data File
EMPIRE STATE BUILDING
Cost
Ticket?
......................... $16.00 Adults (18–61) ............... ........................ $14.00 Youth (12–17).................. ...................... $10.00 Child (6–11) .................... ...................... $14.00 ..... ..... Seniors (62+) .......... ......................... $14.00 Military w/ID ............... ...................... Free Military in Uniform .......... ) ......................... Free Toddlers (5 or younger .....
Source: esbnyc.com
BIGHORN SHEEP
Opening Date: Spring 199 9 Length: 938 feet Height: 125 feet Top Speed: 60 mph Ride Time: 1 minute, 44 seconds Hourly Ride Capacity: 750 passengers per hour Source: sixflags.com
Weight: 115–280 lb Horn Weight: 40–60 lb Length with Tail: 50–62 in. Shoulder Height: 32–40 in. Source: desertusa.com
MINNESOTA VIKINGS
Rushing Statistics Yards Rushed
Average Yards per Rush
Length of Longest Rush
662
4.3
33
155
473
3.8
61
126
147
6.1
18
24
62
1.9
15
32
53
2.9
16
18
28
9.3
11
3
27
6.8
13
4
Number of Rushes
Mewelde Moore Michael Bennett
Player
Daunte Culpepper Ciatrick Fason Brad Johnson Troy Williamson Koren Robinson
The official state mammal of Colorado is the Bighorn Sheep.
Source: espn.com
United States Data File (l)Jeremy Murphy/www.lonestarthrills.com, (r)John Conrad/CORBIS
19
SUNFLOWER
MOUNT RUSHM
OR E
ica.com
Source: britann
er of Kansas is The state flow 1–4.5 cm Stem Height: 7.5–30 cm Leaf Length: 15–30 cm Head Width:
PUBLIC LIBRARY
Source: mountru
shmoreinfo.com
Mount Rushm ore is located in Black Hills, South D akota. • The length of each presid ent’s face is 60 feet from th e top of the h ead to the chin. • A total of 80 0 million pou nds of stone was carved o ut to complete the sculpture. • Each carvin g is scaled to men who would be 465 feet tall. • The monum ent can be seen from 60 miles away .
The Seattle Public Library’s new building, designed by architect Joshua Ramos, opened to the public in 2004. His 362,876 square foot design is comprised of 8 horizontal layers. Each layer varies in size to fit its function. At one corner of the library, the wedge-shaped base nearly diminishes to a point. The total cost of the building was about $1.7 million and its construction required 2050 tons of concrete and 4644 tons of steel.
GREAT LAKES Ontario Quebec
Su pe rio r ron Hu
Michigan
Minnesota Wisconsin
Michigan ie Er
Illinois Indiana
Source: greatbuildings.com
io Ontar
New York
Pennsylvania Ohio Image © GLIN
Source: infoplease.com
20
. the sunflower
United States Data File
(l)Garry Black/Masterfile; (tr)David C. Tomlinson/Getty Images; (br)Ron Wurzer/Getty Images
Lake
Area (mi2)
Erie Huron Michigan Ontario Superior
9940 23,010 22,178 7540 31,010
Maximum Surface Maximum Elevation (ft) Depth (ft) 572 580 581 246 602
210 750 923 778 1302
U.S. CELLULAR FIELD
United States Data File
G R AN
D CAN
YON
Source: ballparks.com
The 40,615-seat U.S. Cellular Field is home to the Chicago White Sox baseball team. The 1,300,00 square foot complex features a 15,000 square foot baseball and softball instruction area for kids. The baseball diamond is a square measuring 90 feet on each side.
The G ra is hom nd Canyon e Natio n seven to the Gra nd Ca al Park in natura A nyon l won one of rizona ders. Lengt t he h Width : 227 miles : Eleva 0.25–15 mil tion: 2 e 400–7 s Source 000 fe : nps.g et ov
TEMPERATURE Wisconsin: Temperature Extremes Month
Maximum °F
Year
Place
Minimum °F
Year
Place
January
66
1897
Prairie DuChien
-54
1922
Danbury
February
69
2000
Afton/Beloit/Broadhead
-55
1996
Couderay
March
86
1986
Dodge
-48
1962
Couderay
April
97
1980
Lone Rock
-20
1924
Rest Lake
May
109
1934
Prairie DuChien
7
1966
Gorden
June
106
1934
Racine
20
1964
Danbury
July
114
1936
Wisconsin Dells
27
1972
Jump River
August
108
1988
University of Wisconsin–Arboret
22
1950
Coddington Exp Farm
September
104
1939
Prairie DuChien
10
1949
Coddington Exp Farm
October
95
1897
Gratiot
-7
1925
Long Lake
November
84
1904
Prairie DuChien
-34
1898
Osceola
December
67
1998
La Crosse
-52
1983
Couderay
Source: infoplease.com
United States Data File (l)Ron Vesely/MLB Photos via Getty Images; (r)J. A. Kraulis/Masterfile
21
Algebra and Integers Focus Build a foundation of basic understandings of numbers, operations, and algebraic thinking. Use these understandings to solve linear equations.
CHAPTER 1 The Tools of Algebra Select and use appropriate operations to solve problems and justify solutions. Use graphs, tables, and algebraic representations to make predictions and solve problems.
CHAPTER 2 Integers Understand that different forms of numbers are appropriate for different situations. Use graphs, tables, and algebraic representations to make predictions and solve problems.
CHAPTER 3 Equations Select and use appropriate operations to solve problems and justify solutions. Use graphs, tables, and algebraic representations to make predictions and solve problems. 22 Unit 1 Algebra and Integers Laurence Parent
Algebra and Geography I Need a Vacation! Does your family like to go camping? Eastern Kentucky is home to the Daniel Boone National Park, which covers over 704,000 acres of rugged terrain. This forest provides excellent opportunities for outdoor recreation. In this project, you will explore how graphs and formulas can help you plan a family vacation. Log on to pre-alg.com to begin.
Unit 1 Algebra and Integers
23
The Tools of Algebra
1 •
Select and use appropriate operations to solve problems and justify solutions
•
Solve problems connected to everyday experiences and other subjects.
Key Vocabulary algebra (p.37) evaluate (p.32) solving the equation (p.49) variable (p.37)
Real-World Link Water Parks You can use the expression 2a + 2c to find price of admission for a family of 2 adults and 2 children to a water park where a is the price of an adult and c is the price of a child.
Problem Solving Make this Foldable to help you organize your strategies for solving problems. Begin with a sheet of paper.
1 Fold the short sides so they meet in the middle.
2 Fold the top to the bottom.
3 Unfold. Cut along the
4 Label each of the
second fold to make four tabs.
tabs as shown.
24 Chapter 1 The Tools of Algebra Rio Aventura, Schlitterbahn Beach Waterpark, South Padre Island, TX
%XPLORE
0LAN
#HECK
3OLVE
GET READY for Chapter 1 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Find each sum or difference. (Prerequisite Skills, p. 745)
1. 6.6 + 8.2 3. 2.65 + 0.3 5. 4.25 - 0.7
2. 5.4 - 2.3 4. 1.08 + 1.2 6. 4.3 - 2.89
7. LUNCH Chad has $8.60. He spends $4.90 on lunch. How much money does Chad have left? (Prerequisite Skills, p. 745)
Example 1
Find 11.9 - 2.15. 8 10
11.9 0 Annex a zero to align the decimal points - 2.1 5 and subtract. 9.7 5 11.9 - 2.15 = 9.75
8. MONEY Jamal has $10.50. His sister Syreeta gives him the $5.75 she borrowed from him yesterday. How much money does Jamal have now? (Prerequisite Skills, p. 745)
Estimate each sum, difference, product, or quotient. (Prerequisite Skill) 9. 328 + 879 10. 22,431 - 13,183 11. 189 × 89 12. 21,789 ÷ 97 13. ANIMALS The peregrine falcon reaches horizontal cruising speeds of 55 miles per hour. At that rate, about how far can it travel in 3 hours? (Prerequisite Skill)
Example 2
Estimate 117 + 51. 117 + 51
→
120 Round to the nearest ten + 50 and add. 170
117 + 51 ≈ 170
Estimate each sum, difference, product, or quotient. (Prerequisite Skills, p. 744) 14. 8.8 + 5.3 15. $7.34 - $2.16 16. 4.2 × 29.3 17. 18.8(5.3) 18. 7.8 ÷ 2.3 19. 21.3 ÷ 1.7
Example 3
20. WEATHER Annual precipitation in Cape Hatteras, North Carolina, is about 56.1 inches. The city of Raleigh receives about 41.4 inches annually. About how much more precipitation does Cape Hatteras receive than Raleigh? (Prerequisite
3.2 × 61.5 ≈ 186
Estimate 3.2 × 61.5. 61.5 × 3.2
→
62 Round to the nearest whole × 3 number and multiply. 186
Skills, p. 744) Chapter 1 Get Ready for Chapter 1
25
1-1
Using a Problem-Solving Plan
Main Ideas • Use a four-step plan to solve problems. • Choose an appropriate method of computation.
New Vocabulary conjecture inductive reasoning
The first postage stamp ever issued by the United States Government was issued on July 1, 1847. The rate for a half-ounce letter was 5¢ for any distance up to 300 miles and 10¢ for any distance over 300 miles. The table shows first-class mail rates in 2006. a. Describe the pattern in the costs. b. How can you determine the cost to mail a 6-ounce letter? c. Suppose you were asked to find the cost of mailing a letter that weighs 8 ounces. What steps would you take to solve the problem?
Weight (oz)
Cost
1
$0.39
2
$0.63
3
$0.87
4
$1.11
5
$1.35
U.S. MAIL
Source: www.usps.gov
Four-Step Problem-Solving Plan It is often helpful to have an organized plan to solve math problems. The following four steps can be used to solve any math problem. 1. Explore
• Read the problem quickly to gain a general understanding of it. • Ask, “What facts do I know?” • Ask, “What do I need to find out?” • Ask, “Is there enough information to solve the problem? Is there extra information?”
2. Plan
• Reread the problem to identify relevant facts. • Determine how the facts relate to one another. • Make a plan and choose a strategy for solving it. There may be several strategies that you can use. • Estimate the answer.
3. Solve
• Use your plan to solve the problem. • If your plan does not work, revise it or make a new plan.
4. Check
• Reread the problem. Is there another solution? • Ask, “Is my answer reasonable and close to my estimate?” • Ask, “Does my answer make sense?” • If not, make a new plan and solve the problem another way.
26 Chapter 1 The Tools of Algebra
POSTAL SERVICE Refer to page 26. How much would it cost to mail a 9-ounce letter first class? Explore The table shows the weight of a letter and the cost to mail it first class. We need to find the cost to mail a 9-ounce letter. Plan
Use the information in the table. Look for a pattern in the costs. Extend the pattern to find the cost for a 9-ounce letter.
Solve
First, find the pattern. Weight (oz)
1
Cost
2
3
$0.39 $0.63 $0.87
4
5
$1.11
$1.35
+ 0.24 + 0.24 + 0.24 + 0.24
Each consecutive cost increases by $0.24. Next, extend the pattern. Weight (oz)
5
Cost
6
7
8
9
$1.35 $1.59 $1.83 $2.07 $2.31 + 0.24 + 0.24 + 0.24 + 0.24
It would cost $2.31 to mail a 9-ounce letter. Reasonableness
Always check to be sure your answer is reasonable. If the answer seems unreasonable, solve the problem again.
Check
To mail a 9-ounce letter, it would cost $0.39 for the first ounce and 8 × $0.24 or $1.92 for the eight additional ounces. Since $0.39 + $1.92 = $2.31, the answer is correct.
1. It costs $0.80 per ounce to mail a letter to England. How much would it cost to mail an 8-ounce letter to England? Personal Tutor at pre-alg.com
A conjecture is an educated guess. When you make a conjecture based on a pattern of examples or past events, you are using inductive reasoning.
EXAMPLE
Use Inductive Reasoning to Solve Problems
a. Find the next term in 1, 3, 9, 27, 81, …. 1
3 ×3
9 ×3
27 ×3
81 ×3
? ×3
Assuming the pattern continues, the next term is 81 × 3 or 243. b. Draw the next figure in the pattern.
The shaded square moves counterclockwise. Assuming the pattern continues, the shaded square will be at the bottom left of the figure.
2. Find the next term in 71, 59, 47, 35, . . . . Extra Examples at pre-alg.com
Lesson 1-1 Using a Problem-Solving Plan
27
Choose the Method of Computation Choosing the method of computation is Look Back
To review problem solving strategies, see pages 2–13.
also an important step in solving problems. In addition to using paper and pencil to solve problems, you can use number sense, estimation, mental math, and a calculator.
Do you need an exact answer?
Do you see a pattern or number fact?
yes
yes
Use mental math.
yes
Use paper and pencil.
no no Estimate.
Use a calculator.
no
Are there simple calculations to do?
Explore
You know the seating capacities of Comerica Park and Fenway Park. You need to find how many more seats Comerica Park has than Fenway Park.
Plan
The question uses the word about, so an exact answer is not needed. We can solve the problem using estimation.
Real-World Link Fenway Park was built in 1912 and is the oldest stadium in Major League Baseball. Source: Major League Baseball
Solve
40,950 40,000
TICKET
41,503 TICKET
TICKET
39,345 TICKET
33,871 TICKET
Pa Come rk (De rica Fe tro nw it) ay Pa rk (Bo sto n) Mi Pa nute rk (Ho Maid ust on ) (Sa SB C nF P ran ark Wr cis igl co) ey Fie ld (Ch ica go )
BASEBALL The graph shows the seating capacity of certain baseball stadiums. About how many more seats does Comerica Park have than Fenway Park?
Seating capacity
Ballpark Seating Capacity 42,000 41,000 40,000 39,000 38,000 37,000 36,000 35,000 34,000 33,000 32,000 31,000 30,000 0
Ballpark Source: Major League Baseball
Comerica Park: 40,000 → 40,000 Round to the nearest thousand. Fenway Park: 33,871 → 34,000 40,000 - 34,000 = 6000 Subtract 34,000 from 40,000. So, Comerica Park has about 6000 more seats than Fenway Park.
Check
Add 33,871 and 6000. Since 39,871 ≈ 40,000, the answer is reasonable.
3. About how many more seats does Minute Maid Park have than Wrigley Field? 28 Chapter 1 The Tools of Algebra Icon SMI/CORBIS
Example 1 (p. 27)
Example 2 (p. 27)
1. TIME The ferry schedule shows that the ferry departs at regular intervals. Use the four-step plan to find the earliest time Brady can catch the ferry if he cannot leave until 1:30 P.M.
South Bass Island Ferry Schedule
Find the next term in each list. 2. 10, 20, 30, 40, 50, …
Departures
Arrivals
8 : 4 5 A.M.
9:
9 : 3 3 A.M.
1
10:21 A.M.
3. 37, 33, 29, 25, 21, …
11:09 A.M.
4. 12, 17, 22, 27, 32, … 5. 3, 12, 48, 192, 768, … GEOMETRY Draw the next pattern in the figure. 6.
Example 3 (p. 28)
HOMEWORK
HELP
For See Exercises Examples 8–9 1 10–19 2 20–21 3
7. MONEY In 2003, the average U.S. household spent $13,432 on housing, $2060 on entertainment, $5340 on food, and $7781 on transportation. How much was spent on housing each month? Round to the nearest cent.
ANALYZE TABLES For Exercises 8 and 9, use the table that gives the approximate heart rate for a person exercising at 85% intensity. Age
20
25
30
35
40
45
Heart Rate (beats/min)
174
170
166
162
158
154
8. Assume the pattern continues. Use the four-step plan to find the heart rate a 15-year-old should maintain while exercising at this intensity. 9. What heart rate should a 55-year-old maintain while exercising at this intensity? Find the next term in each list. 10. 2, 5, 8, 11, 14, …
11. 4, 8, 12, 16, 20, …
12. 0, 5, 10, 15, 20, …
13. 2, 6, 18, 54, 162, …
14. 54, 50, 46, 42, 38, …
15. 67, 61, 55, 49, 43, …
16. 2, 5, 9, 14, 20, …
17. 3, 5, 9, 15, 23, …
GEOMETRY Draw the next figure in each pattern. 18. 19.
20. SAVINGS Juan needs to save $125 for a ski trip. He has $68 in his bank. He receives $15 for an allowance and earns $20 delivering newspapers and $16 shoveling snow. Does he have enough money for the trip? Explain. 21. COINS Using eight coins, how can you make change for 65 cents that will not make change for a quarter? Lesson 1-1 Using a Problem-Solving Plan
29
22. MEDICINE The numbers of different types of transplants that were performed in the United States in a recent year are shown in the table. About how many transplants were performed?
EXTRA
PRACTIICE
See pages 761, 794. Self-Check Quiz at pre-alg.com
H.O.T. Problems
23. CANDY A gourmet jelly bean company can produce 100,000 pounds of jelly beans a day. One ounce of these jelly beans contains 100 Calories. If there are 800 jelly beans in a pound, how many jelly beans can be produced in a day?
Transplant Heart Heart-Lung Intestine Kidney Kidney-Pancreas Liver Lung Pancreas
Number 2155 33 107 14,775 905 5329 1042 554
Source: The World Almanac
24. WATER A water tank is draining at a rate of 12 gallons every 8 minutes. If there are 234 gallons in the tank, when will it have just 138 gallons left? 25. OPEN ENDED Write a list of numbers in which 4 is added to get each succeeding term. CHALLENGE For Exercises 26 and 27, think of a 1-to-9 multiplication table. 26. Are there more odd or more even products? How can you determine the answer without counting? 27. Is this different from a 1-to-9 addition facts table? Explain. 28.
Writing in Math Explain why it is helpful to use a plan to solve problems. Include an explanation of the importance of performing each step of the four-step problem-solving plan.
29. What is the relationship between the number of cuts and the number of pieces in each circle?
A The number of pieces is half the number of cuts. B The number of pieces is the same as the number of cuts.
30. Suppose you had 1200 sugar cubes. What is the largest cube you could build with the sugar cubes? F 8 by 8 by 8
H 11 by 11 by 11
G 10 by 10 by 10
J 12 by 12 by 12
31. Antonia bought a video game system for $323.96. She paid in 12 equal installments. Which is the best estimate for the amount of each payment?
C The number of cuts is twice the number of pieces.
A less than $20 B between $20 and $25
D The number of cuts is half the number of pieces.
C between $25 and $30 D greater than $30
PREREQUISITE SKILL Round each number to the nearest whole number. (p. 743) 32. 2.8
33. 5.2
30 Chapter 1 The Tools of Algebra
34. 35.4
35. 49.6
36. 109.3
Translating Expressions into Words Chinese, English, French, Russian, Spanish, and Arabic are the official languages of the United Nations. All formal meetings and all official documents, in print or online, are interpreted in all six languages. Translating numerical expressions into verbal phrases is an important skill in algebra. Key words and phrases play an essential role in this skill. The following table lists some words and phrases that suggest addition, subtraction, multiplication, and division. Addition
Subtraction
plus sum more than increased by in all
minus difference less than subtract decreased by less
Multiplication times product multiplied each of factors
Division divided quotient per rate ratio separate
A few examples of how to write an expression as a verbal phrase are shown. Expression 5×8 2+4 16 ÷ 2 8-6 2×5 5-2
Key Word times sum quotient less than product less
Verbal Phrase 5 times 8 the sum of 2 and 4 the quotient of 16 and 2 6 less than 8 the product of 2 and 5 5 less 2
Reading to Learn 1. Refer to the table above. Write a different verbal phrase for each expression. Choose the letter of the phrase that best matches each expression. 2. 9 - 3 a. the sum of 3 and 9 3. 3 ÷ 9 b. the quotient of 9 and 3 4. 9 · 3 c. 3 less than 9 5. 3 + 9 d. 9 multiplied by 3 6. 9 ÷ 3 e. 3 divided by 9 Write two verbal phrases for each expression. 7. 5 + 1 8. 8 + 6 9. 9 × 5 11. 12 ÷ 3
12. 20 4
13. 8 - 7
10. 2(4) 14. 11 - 5 Reading Math Translating Expressions into Words
Mike Segar/Reuters/CORBIS
31
1-2
Numbers and Expressions
Main Ideas • Use the order of operations to evaluate expressions. • Translate verbal phrases into numerical expressions.
New Vocabulary numerical expression evaluate order of operations
Scientific calculators are programmed to find the value of an expression in a certain order. Expression Value
1+2×5
8-4÷2
10 ÷ 5 + 14 × 2
11
6
30
a. Study the expressions and their respective values. For each expression, tell the order in which the calculator performed the operations. b. For each expression, does the calculator perform the operations in order from left to right? c. Based on your answer to parts a and b, find the value of each expression below. Check your answer with a scientific calculator. 12 - 3 × 2 16 ÷ 4 - 2 18 + 6 - 8 ÷ 2 × 3 d. Make a conjecture as to the order in which a scientific calculator performs operations.
Order of Operations Expressions like 1 + 2 × 5 and 10 ÷ 5 + 14 ÷ 2 are numerical expressions. Numerical expressions contain numbers and operations such as addition, subtraction, multiplication, and division. When you evaluate an expression, you find its numerical value. To avoid confusion, mathematicians have agreed upon the following order of operations. Order of operations are rules to follow when more than one operation is used in an expression. Order of Operations
Reading Math Grouping Symbols Grouping symbols include: • parentheses ( ), • brackets [ ], and • fraction bars, as in 6+4 _ , which means 2 (6 + 4) ÷ 2.
Step 1 Evaluate the expressions inside grouping symbols. Step 2 Multiply and/or divide in order from left to right. Step 3 Add and/or subtract in order from left to right.
Numerical expressions have only one value. Consider 6 + 4 × 3. 6 + 4 × 3 = 6 + 12 = 18
Multiply, then add.
6 + 4 × 3 = 10 × 3 = 30
Add, then multiply.
Using the order of operations, the correct value of 6 + 4 × 3 is 18. 32 Chapter 1 The Tools of Algebra
EXAMPLE
Evaluate Expressions
Find the value of each expression. a. 18 ÷ 3 × 2 18 ÷ 3 × 2 = 6 × 2 Divide 18 by 3. = 12
Reading Math Multiplication and Division Notation
A raised dot or parentheses represents multiplication. A fraction bar represents division.
Multiply 6 and 2.
b. 6(2 + 9) - 3 · 8 6(2 + 9) - 3 · 8 = 6(11) - 3 · 8
Evaluate (2 + 9) first.
= 66 - 3 · 8
6(11) means 6 × 11.
= 66 - 24
3 · 8 means 3 times 8.
= 42
Subtract 24 from 66.
c. 4[(15 - 9) + 8(2)] 4[(15 - 9) + 8(2)] = 4[6 + 8(2)]
Evaluate (15 - 9).
= 4(6 + 16)
Multiply 8 and 2.
= 4(22)
Add 6 and 16.
= 88
Multiply 4 and 22.
53 + 15 d. _ 17 - 13 53 + 15 _ = (53 + 15) ÷ (17 - 13) 17 - 13
Rewrite as a division expression.
= 68 ÷ 4
Evaluate 53 + 15 and 17 - 13.
= 17
Divide 68 by 4.
1A. 6 - 3 + 5
1B. 24 ÷ 3 × 9
1C. 2[(10 - 3) + 6(5)]
19 - 7 1D. _ 25 - 22
Personal Tutor at pre-alg.com
Translate Verbal Phrases into Numerical Expressions You have learned to translate numerical expressions into verbal phrases. It is often necessary to translate verbal phrases into numerical expressions.
EXAMPLE
Translate Phrases into Expressions
Reading Math
Write a numerical expression for each verbal phrase.
Differences and Quotients In this book,
a. the product of eight and seven
the difference of 9 and 3 means to start with 9 and subtract 3, so the expression is 9 - 3. Similarly, the quotient of 9 and 3 means to start with 9 and divide by 3, so the expression is 9 ÷ 3.
Words Expression
the product of eight and seven 8×7
b. the difference of nine and three Words Expression
the difference of nine and three 9-3
2A. the sum of 10 and 3 Extra Examples at pre-alg.com
2B. the quotient of 14 and 7 Lesson 1-2 Numbers and Expressions
33
CELL PHONES A cell phone company charges $20 per month and $0.10 for each call made or received. Write and then evaluate an expression to find the cost for 44 calls during one month. Words
and
Expression
+
20 + 0.10 × 44 = 20 + 4.40 = $24.40
f
ll × 44
Multiply. Add.
3. MONEY A taxi charges $4 for the first mile and $2 for each additional mile. Write and evaluate an expression for the fare for a 10-mile trip.
Example 1 (p. 33)
Example 2 (p. 33)
Example 3 (p. 34)
HOMEWORK
HELP
For See Exercises Examples 15–26 1 27–34 2 35–37 3
Find the value of each expression. 1. 32 - 24 ÷ 2
2. 18 + 2 × 4
3. 2 × 9 ÷ 3
4. 5(8) + 7
5. 6(15 - 4)
6. 2[3 + 7(4)]
7. 3[(20 - 7) + 1]
10 - 4 8. _ 1+2
34 + 18 9. _ 27 - 14
Write a numerical expression for each verbal phrase. 10. the quotient of fifteen and five 11. the product of six and eight 12. the difference of twelve and nine 13. the sum of eleven and sixteen 14. MUSIC Tyler purchased 3 CDs for $13 each and 2 digital songs for $0.99 each. Write and then evaluate an expression for the total cost.
Find the value of each expression. 15. 3 · 6 - 4
16. 12 - 3 × 3
17. 12 ÷ 3 + 21
18. 9 + 18 ÷ 3
19. 8 + 5(6)
20. 12(11) - 56
15 + 9 21. _ 32 - 20
45 - 18 22. _ 9÷3
23. 11(6 - 1)
24. (9 - 7) · 13
25. 56 ÷ (7 · 2) × 6
26. 75 ÷ (7 + 8) - 3
Write a numerical expression for each verbal phrase. 27. 29. 31. 33. 34.
seven increased by two 28. six minus three nine multiplied by five 30. eleven more than fifteen twenty-four divided by six 32. four less than eighteen the cost of 3 notebooks at $6 each the total amount of CDs if Sancho has 4 and Brianna has 5
34 Chapter 1 The Tools of Algebra
ZOO For Exercises 35 and 36, use the information in the table about the price of admission to a zoo. 35. Write an expression that can be used to find the total cost of admission for 4 adults, 3 children, and 1 senior. 36. Find the total cost.
:OO !DMISSION 4ICKET !DULTS #HILDREN 3ENIORS
#OST
37. TRAVEL Joshua is packing for a trip. The total weight of his luggage cannot exceed 70 pounds. He has 3 suitcases that weigh 16 pounds each and 2 sport bags that weigh 9 pounds each. Is Joshua’s luggage within the 70-pound limit? Explain your reasoning. ANALYZING TABLES For Exercises 38 and 39, use the table and the following information. A national poll ranks college football teams using votes from sports reporters. Each vote is worth a certain number of points. Suppose that Penn State University receives 50 first-place votes, 7 second-place votes, 4 fourth-place votes, and 3 tenth-place votes. 38. Write an expression for the number of points that Penn State receives. 39. Find the total number of points.
Number of Points for Each Vote Vote
Points
1st place
25
2nd place
24
3rd place
23
4th place
22
5th place
21
25th place
1
40. Find the value of six added to the product of four and eleven. 41. What is the value of sixty divided by the sum of two and ten? Copy each sentence. Then insert parentheses to make each sentence true.
EXTRA
PRACTICE
See pages 761, 794. Self-Check Quiz at pre-alg.com
H.O.T. Problems
42. 61 - 15 + 3 = 43
43. 12 × 3 ÷ 1 + 2 = 12
44. 56 ÷ 2 + 6 - 4 = 3
45. 5 + 2 · 9 - 3 = 42
46. PUBLISHING An International Standard Book Number (ISBN) is used to identify a published book. To determine if an ISBN is correct, multiply each digit in order by 10, 9, 8, 7, and so on. If the sum of the products can be divided by 11, with no remainder, the number is correct. Find the 10-digit ISBN on the back cover of this book. Is the number correct? Explain why or why not. 47. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would evaluate an expression. 48. OPEN ENDED Give an example of an expression involving multiplication and subtraction in which you would subtract first. 49. FIND THE ERROR Emily and Marcus are evaluating 24 ÷ 2 × 3. Who is correct? Explain your reasoning. Emily 24 ÷ 2 x 3 = 12 x 3 = 36
Marcus 24 ÷ 2 × 3 = 24 ÷ 6 =4
50. REASONING Do 2 × 4 + 3 and 2 × (4 + 3) have the same value? Explain. Lesson 1-2 Numbers and Expressions
35
51. CHALLENGE Suppose only the 1, , , ⫻ , ⫼ , ( , ) , and ENTER keys on a calculator are working. How can you get a result of 75 if you are only allowed to push these keys fewer than 20 times? 52.
Writing in Math
Explain why there should be an agreement on the
order of operations.
53. A bag of potting soil sells for $2, and a bag of fertilizer sells for $13. What is the expression for the total cost of 4 bags of soil and 2 bags of fertilizer? A (4 × 2) + (2 × 13)
54. GRIDDABLE The final standings of a hockey league are shown. A win is worth three points, and a tie is worth 1 point. Zero points are given for a loss. How many points did the Wildcats have?
B (4 × 13) + (2 × 2) C 4(2 + 13) D (2 + 13)(4 + 2)
Team
Wins
Losses
Ties
Knights Huskies Wildcats Mustangs Panthers
14 11 10 9 10
9 9 9 10 14
7 10 11 11 6
Find the next term in each list. (Lesson 1-1) 55. 2, 4, 8, 16, 32, . . .
56. 45, 42, 39, 36, 33, . . .
57. 20, 33, 46, 59, 72, . . .
58. 1, 3, 6, 10, 15, 21, . . .
59. 15, 18, 22, 25, 29, . . .
60. 1215, 405, 135, 45, . . .
Solve each problem. (Lesson 1-1) 61. BUSINESS Mr. Armas is a sales associate for a computer company. He receives a salary plus a bonus for any computer package he sells. Find Mr. Armas’ bonus if he sells 16 computer packages. 62. SPACE SHUTTLE The space shuttle can carry a payload of about 65,000 pounds. If a compact car weighs about 2450 pounds, about how many compact cars can the space shuttle carry? 63. TRAVEL The graph shows the number of travelers to the top five tourist destinations in a recent year. About how many more people traveled to France than to the United States?
Packages
Bonus
2
$100
4
$125
6
$150
8
$175
Top Five Tourism Destinations 90
Arrivals (millions)
76.5 70
49.5
50
45.5 39.1 33.2
PREREQUISITE SKILL Find each sum. 64. 18 + 34
65. 85 + 41
66. 78 + 592
67. 35 + 461
30 0 France
United States
Country Source: infoplease.com
36 Chapter 1 The Tools of Algebra
Spain
Italy
China
1-3
Variables and Expressions
algebra variable algebraic expression defining a variable
Money Earned
2 5 8 11 h
5 · 2 or 10 5 · 5 or 25 5 · 8 or 40 5 · 11 or 55 ?
a. How much would the baby-sitter earn for working 10 hours?
5
5
New Vocabulary
Number of Hours
F
• Translate verbal phrases into algebraic expressions.
A baby-sitter earns $5 per hour. The table shows several possibilities for number of hours and earnings.
F
• Evaluate expressions containing variables.
5
Main Ideas
5
b. What is the relationship between the number of hours and earnings? c. If h represents any number of hours, what expression could you write to represent the amount of money earned?
Evaluate Expressions Algebra is a branch of mathematics dealing with symbols. One symbol that is frequently used is a variable. A variable is a placeholder for any value. As shown above, h represents some unknown number of hours. Any letter can be used as a variable. Notice the special notation for multiplication and division with variables. The letter x is most often used as a variable.
x+2
_y means y ÷ 3.
4h means 4 × h. mn means m × n.
4h - 5
3
mn
y 3
An expression like x + 2 is an algebraic expression because it contains sums and/or products of variables and numbers. To evaluate an algebraic expression, replace the variable or variables with known values and then use the order of operations.
EXAMPLE
Evaluate Expressions
Evaluate x + y - 9 if x = 15 and y = 26. x + y - 9 = 15 + 26 - 9
Replace x with 15 and y with 26.
= 41 - 9
Add 15 and 26.
= 32
Subtract 9 from 41.
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
1A. Evaluate 6 - e + f if e = 3 and f = 9. 1B. Evaluate 7k + h if k = 4 and h = 10. Lesson 1-3 Variables and Expressions
37
EXAMPLE
Evaluate Expressions
Evaluate each expression if k = 2, m = 7, and n = 4. a. 6m - 3k 6m - 3k = 6(7) - 3(2) = 42 - 6 or 36
Replace m with 7 and k with 2. Multiply. Then subtract.
b. mn 2 mn = mn ÷ 2 2
Rewrite as a division expression.
= (7 · 4) ÷ 2
Replace m with 7 and n with 4.
= 28 ÷ 2 or 14 Multiply. Then divide. c. n (k 5m) n + (k + 5m) = 4 + (2 + 5 · 7)
Replace n with 4, k with 2, and m with 7.
= 4 + (2 + 35)
Multiply 5 and 7.
= 4 + 37 or 41
Add 2 and 35. Then add 4 and 37.
Evaluate each expression if r = 1, s = 5, and t = 8. st 2A. 6s + 2r 2B. _ 20
2C. r + (40 - 3t)
Translate Verbal Phrases The first step in translating verbal phrases into algebraic expressions is to choose a variable and a quantity for the variable to represent. This is called defining a variable.
EXAMPLE Vocabulary Link Variable Everyday Use likely to change or vary Math Use a letter representing a value that can vary
Translate Verbal Phrases into Expressions
Translate each phrase into an algebraic expression. a. twelve points more than the Falcons scored Words
twelve points more than the Falcons scored.
Variable
Let p represent the points the Falcons scored.
Expression
p + 12
b. four times a number decreased by 6 Words Variable
four times a number decreased by 6 Let n represent the number.
Expression 4n
3A. two miles less than the athlete ran 3B. five more than three times a number 38 Chapter 1 The Tools of Algebra
Sometimes problems include more than one unknown quantity. You must define a variable for each unknown. Then you can write an expression to represent the situation.
SOCCER The Johnstown Soccer League ranks each team in its league using points. A team gets three points for a win and one point for a tie. a. Write an expression that can be used to find the total number of points a team receives. Words
three points times number of wins plus one point times numbers of ties
Variables
Let w = number of wins and t = number of ties.
Expression 3w + 1t
The expression is 3w + 1t or 3w + t. b. Suppose in one season, the North Rockets had 17 wins and 4 ties. How many points did they receive? 3w + 1t = 3(17) + 1(4)
Real-World Link Soccer is the most popular sport in the world. It is estimated that more than 240,000,000 people play soccer around the world. Source: The World Almanac for Kids
Replace w with 17 and t with 4.
= 51 + 4
Multiply.
= 55
Then add.
4. PHOTOGRAPHY A studio charges a sitting fee of $25 plus $4 for each 4-inch by 6-inch print. Write an expression that can be used to find the total cost to have photographs taken. Then find the cost of purchasing twelve 4-inch by 6-inch prints. Personal Tutor at pre-alg.com
Examples 1, 2 (pp. 37–38)
Example 3 (pp. 38)
Example 4 (pp. 39)
ALGEBRA Evaluate each expression if a = 5, b = 12, and c = 4. 1. b + 6
2. a - 3
3. 20 - c + a
4. 18 - 3c
2b 5. _ 8
6. 5a - (b - c)
ALGEBRA Translate each phrase into an algebraic expression. 7. 8. 9. 10.
eight dollars more than the amount Taimi saved five goals less than the Pirates scored the quotient of a number and four, minus five seven increased by the quotient of a number and eight
CAPACITY For Exercises 11 and 12, use the following information. One pint of liquid is the same as 16 fluid ounces. 11. Suppose the number of pints of liquid is represented by p. Write an expression to find the number of fluid ounces. 12. How many fluid ounces is 5 pints?
Extra Examples at pre-alg.com Najlah Feanny/CORBIS
Lesson 1-3 Variables and Expressions
39
HOMEWORK
HELP
For See Exercises Examples 13–14 1 15–24 2 25–30 3 31–34 4
ALGEBRA Evaluate each expression if x = 7, y = 3, and z = 9. 13. z + 2
14. 5 + x
15. 2 + 4z
16. 15 - 2x
6y 17. _ z
19. 3x - 2y
20. 4z - 3y
xz 21. 10 - _
23. 2x + 3z + 5y
24. 5z - 3x - 2y
xy 22. _ + 2 3
9x 18. _ y
9
ALGEBRA Translate each phrase into an algebraic expression. 25. Bianca’s salary plus a $200 bonus 26. three more than the number of cakes baked 27. six feet shorter than the mountain’s height 28. two seconds slower than Joseph’s time 29. three times as many balloons 30. the product of 12 and a number SCIENCE For Exercises 31 and 32, use the following information. The number of times a cricket chirps can be used to estimate the temperature in degrees Fahrenheit. Use c ÷ 4 + 37, where c is the number of chirps in 1 minute. 31. Find the approximate temperature if a cricket chirps 136 times in a minute. 32. What is the temperature if a cricket chirps 100 times in a minute?
Real-World Link
SHOPPING For Exercises 33 and 34, use the following information. The selling price of a sweater is the cost of the sweater plus the markup minus the discount. 33. Write an expression to show the selling price s of a sweater. Use c for cost, m for markup, and d for discount. 34. Suppose the cost of a sweater is $25, the markup is $20, and the discount is $6. What is the selling price of the sweater?
To convert cricket chirps to degrees Celsius, count the number of chirps in 25 seconds, divide by 3, then add 4 to get temperature.
ALGEBRA Evaluate each expression if x 9, y 4, and z 12.
Source: almanac.com
ANALYZE TABLES Write an algebraic expression that represents the relationship in each table.
35. 7z - (y + x)
36. (8y + 5) - 2z
37. (5z - 4x) + 3y
38. 6x - (z - 2y)
39. 2x + (4z - 13) - 5
40. (29 - 3y) + 4z - 7
41.
EXTRA
PRACTIICE
See pages 761, 794. Self-Check Quiz at pre-alg.com
Age in Three Years
10
13
5
12
15
of Items
Total Cost
43. Regular Price
Sale Price
$25
$12
6
$30
$15
$11
$40
$18
$14
$8
15
18
8
20
23
10
$50
$24
$20
x
?
n
?
$p
?
ALGEBRA Translate each phrase into an algebraic expression. 44. seven less than the product of a number and eight 45. twice a number decreased by the quotient of eight and twice the number
40 Chapter 1 The Tools of Algebra Dennis Johnson/Papilio/CORBIS
42. Number
Age Now
H.O.T. Problems
46. OPEN ENDED Give two examples of algebraic expressions. Then give two examples of expressions that are not algebraic. 47. Which One Doesn’t Belong? Suppose a = 2 and b = 5. Identify the expression that does not belong with the other three. Explain your reasoning. a + 3b
6a – b
4b – (a + 1)
12 + b
48. CHALLENGE What value of t makes the expressions 6t, t + 5, and 2t + 4 equal?
Writing in Math Explain how variables are used to show relationships. Include an example to illustrate your reasoning.
49.
50. After the included minutes have been exhausted, a cell phone company charges an additional $0.08 per minute. Plan A uses a flat rate of $0.10 per minute. Which plan is the least costly if a person uses 750 minutes per month? Plan
Monthly Fee
Included Minutes
A B C D
$0 $29.99 $39.99 $49.99
None 500 1,000 1,500
A Plan A
C Plan C
B Plan B
D Plan D
51. Suppose Benito is selling 10 of his music CDs on the Internet, and it costs $1.25 per CD to send them to a buyer. If he decides to sell each CD for the same price p, which expression would you use to find how much money he will receive after sending all 10 CDs? F 22.75p G 22.5p 10 H 10 1.25 J 10p 12.5
Find the value of each expression. (Lesson 1-2) 52. 3 + (6 × 2) - 8
53. 5(16 - 5 × 3)
8 ÷ 8 + 11 54. _
55. 36 ÷ (9 · 2) + 7
56. 70 - (16 ÷ 2 + 21)
57. 4(20 - 13) + 4 × 5
15 - 4(3)
58. FOOD The table shows the amount in pounds of certain types of pasta sold in a recent year. About how many million pounds of these types of pasta were sold? (Lesson 1-1)
59. ANIMALS A Beluga whale’s heart beats about 16 times per minute. Find the number of times a Beluga whale’s heart beats in one hour. (Lesson 1-1)
PREREQUISITE SKILL Find each difference. 60. 53 - 17
61. 97 - 28
62. 104 - 82
Pasta
Amount (millions)
Spaghetti
308
Elbow
121
Noodles
70
Twirl
52
Penne
51
Lasagna
35
Fettuccine
24
Source: National Pasta Association
63. 152 - 123 Lesson 1-3 Variables and Expressions
41
EXTEND
Spreadsheet Lab
1-3
Expressions and Spreadsheets
One of the most common computer applications is a spreadsheet program. A spreadsheet is a table that performs calculations. It is organized into boxes called cells, which are named by a letter and a number. In the spreadsheet below, cell B2 is highlighted. An advantage of using a spreadsheet is that values in the spreadsheet are recalculated when a number is changed. You can use a spreadsheet to investigate patterns in data.
EXAMPLE
Interactive Lab pre-alg.com
Here’s a mind-reading trick! Think of a number. Then double it, add six, divide by two, and subtract the original number. What is the result? You can use a spreadsheet to test different numbers. Suppose we start with the number 10.
-IND 2EADING 4RICK !
"
4HINK OF A NUMBER $OUBLE IT !DD $IVIDE BY
3UBTRACT THE ORIGINAL NUMBER
" " "
" "
3HEET
3HEET
#
3HEET
4HE SPREADSHEET TAKES THE VALUE IN " DOUBLES IT AND ENTERS THE VALUE IN " .OTE THE IS THE SYMBOL FOR MULTIPLICATION 4HE SPREADSHEET TAKES THE VALUE IN " DIVIDES BY AND ENTERS THE VALUE IN " .OTE THAT IS THE SYMBOL FOR DIVISION
The result is 3.
EXERCISES To change information in a spreadsheet, move the cursor to the cell you want to access and click the mouse. Then type in the information and press Enter. Find the result when each value is entered in B1. 1. 6 2. 8 3. 25 4. 100 5. 1500 6. MAKE A CONJECTURE What is the result if a decimal is entered in B1? a negative number? 7. Explain why the result is always 3. Write an expression that describes your answer. 8. Make up your own mind-reading trick. Enter it into a spreadsheet to show that it works. Write an expression to describe the trick. 42 Chapter 1 The Tools of Algebra
1-4
Properties
Main Ideas • Identify and use properties of addition and multiplication. • Use properties of addition and multiplication to simplify algebraic expressions.
Abraham Lincoln delivered the Gettysburg Address more than 130 years ago. The table lists the number of words in certain historic documents.
Historical Document
Words
Preamble to The U.S.Constitution
52
Mayflower Compact
196
Atlantic Charter
375
Gettysburg Address (Nicolay Version)
238
Source: U.S. Historical Documents Archive
New Vocabulary properties counterexample simplify deductive reasoning
a. Suppose you read the Preamble to The U.S. Constitution first and then the Gettysburg Address. Write an expression for the total number of words read. b. Suppose you read the Gettysburg Address first and then the Preamble to the U.S. Constitution. Write an expression for the total number of words read. c. Find the value of each expression. What do you observe? d. Does it matter in which order you add any two numbers? Why or why not?
Vocabulary Link Commute Everyday Use to change or exchange Commutative Math Use property that allows you to change the order in which numbers are added or multiplied
Properties of Addition and Multiplication In algebra, properties are statements that are true for any numbers. For example, the expressions 3 + 8 and 8 + 3 have the same value, 11. This illustrates the Commutative Property of Addition. Likewise, 3 8 and 8 3 have the same value, 24. This illustrates the Commutative Property of Multiplication.
Commutative Property of Addition Words
The order in which numbers are added does not change the sum.
Symbols
For any numbers a and b, a + b = b + a.
Example
2+3=3+2 5=5
Commutative Property of Multiplication Words
The order in which numbers are multiplied does not change the product.
Symbols
For any numbers a and b, a · b = b · a.
Example
2·3=3·2 6=6 Lesson 1-4 Properties
43
When evaluating expressions, it is often helpful to group or associate the numbers. The Associative Property says that the way in which numbers are grouped when added or multiplied does not change the sum or the product. Associative Property of Addition
Vocabulary Link Associate Everyday Use to join together, connect, or combine Associative Math Use property that allows you to change the groupings in which numbers are added or multiplied
Words
The way in which numbers are grouped when added does not change the sum.
Symbols
For any numbers a, b, and c, (a + b) + c = a + (b + c).
Example
(5 + 8) + 2 = 5 + (8 + 2) 13 + 2 = 5 + 10 15 = 15 Associative Property of Multiplication
Words
The way in which numbers are grouped when multiplied does not change the product.
Symbols
For any numbers a, b, and c, (a · b) · c = a · (b · c).
Example
(4 · 6) · 3 = 4 · (6 · 3) 24 · 3 = 4 · 18 72 = 72
The following properties are also true. Properties of Numbers Property
Words
Symbols
Examples
Additive Identity
When 0 is added For any number a, to any number, the a + 0 = 0 + a = a. sum is the number.
5+0=5 0+9=9
Multiplicative Identity
When any number is multiplied by 1, the product is the number.
For any number a, a · 1 = 1 · a = a.
7·1=7 1·6=6
Multiplicative Property of Zero
When any number is multiplied by 0, the product is 0.
For any number a, a · 0 = 0 · a = 0.
4·0=0 0·2=0
EXAMPLE
Identify Properties
Name the property shown by each statement. a. 3 + 7 + 9 = 7 + 3 + 9 The order of the numbers changed. This is the Commutative Property of Addition.
b. (a · 6) · 5 = a · (6 · 5) The grouping of the numbers and variables changed. This is the Associative Property of Multiplication.
1A. 5 × 7 × 2 = 7 × 2 × 5
1B. 14 + (9 + 10) = (14 + 9) + 10
1C. 8 · 1 = 8
1D. 0 · 12 = 0
44 Chapter 1 The Tools of Algebra
EXAMPLE Mental Math
Look for sums or products that end in zero.
Mental Math
Find 4 · (25 · 11) mentally. Group 4 and 25 because 4 · 25 = 100. It is easy to multiply by 100 mentally. 4 · (25 · 11) = (4 · 25) · 11 Associative Property of Multiplication = 100 · 11
Multiply 4 and 25 mentally.
= 1100
Multiply 100 and 11 mentally.
Find each sum or product mentally. 2B. (97 + 25) + 3
2A. 40 · (6 · 5)
Counterexample
You can disprove a statement by finding only one counterexample.
One way to find out if these properties apply to subtraction is to look for a counterexample. A counterexample is an example that shows a conjecture is not true.
EXAMPLE
Find a Counterexample
Is subtraction of whole numbers associative? If not, give a counterexample. Write two subtraction expressions using the Associative Property, and then check to see whether they are equal. 9 - (5 - 3) (9 - 5) - 3 9-24-3 7≠1
State the conjecture. Simplify within the parentheses. Subtract.
We found a counterexample. So, subtraction is not associative.
3. Is subtraction of decimals associative? If not, give a counterexample. Personal Tutor at pre-alg.com
Simplify Algebraic Expressions To simplify algebraic expressions means to write them in a simpler form.
EXAMPLE
Simplify Algebraic Expressions
Simplify each expression. a. (k + 2) + 7
b.
(k + 2) + 7 = k + (2 + 7)
5 · (d · 9) 5 · (d · 9) = 5 · (9 · d)
=k+9
= (5 · 9)d = 45d
4A. 12 · (10 · z)
4B. 10 + (p + 18)
Using facts, properties, or rules to reach valid conclusions is called deductive reasoning. Extra Examples at pre-alg.com
Lesson 1-4 Properties
45
Example 1 (p. 44)
Example 2 (p. 45)
Name the property shown by each statement. 1. 7 + 5 = 5 + 7
2. 8 + 0 = 8
3. 8 · 4 · 13 = 4 · 8 · 13
4. 1 × 6 = 6
5. 13 × 12 = 12 × 13
6. 6 + (1 + 9) = (6 + 1) + 9
MENTAL MATH Find each sum or product. Explain your reasoning. 7. 13 + 8 + 7
8. 6 · 9 · 5
9. 8 + 11 + 22 + 4
10. Is division of whole numbers commutative? If not, give a counterexample. Example 3 (p. 45)
Example 4 (p. 45)
HOMEWORK
HELP
For See Exercises Examples 16–25 1 26–33 2 34–37 3 38–46 4
ALGEBRA Simplify each expression. 11. 6 + (n + 7)
12. (3 + k) + 8
13. (3 · w) · 9
14. 10 · (r · 5)
15. SHOPPING Clara purchased a pair of jeans for $26, a T-shirt for $12, and a pair of socks for $4. What is the total cost of the items without tax? Explain how the Commutative Property of Addition can be used to find the total mentally.
Name the property shown by each statement. 16. 5 · 3 = 3 · 5
17. 12 · 8 = 8 · 12
18. 6 · 2 · 0 = 0
19. 1 · 4 = 4
20. 0 + 13 = 13 + 0
21. (4 + 5) + 15 = 4 + (5 + 15)
22. 1h = h
23. 7k + 0 = 7k
24. (5 + x) + 6 = 5 + (x + 6)
25. 4(mn) = (4m)(n)
MENTAL MATH Find each sum or product. Explain your reasoning. 26. 11 + 8 + 19
27. 17 + 5 + 33
28. 11 · 9 · 10
29. 2 · 7 · 30
30. 15 · 0 · 2
31. 125 · 4 · 0
32. 74 + 22 + 6
33. 23 + 8 + 27
State whether each conjecture is true. If not, give a counterexample. 34. Division of whole numbers is associative. 35. Subtraction of whole numbers is commutative. 36. The sum of two whole numbers is always greater than either addend. 37. The sum of two odd numbers is always odd. ALGEBRA Simplify each expression. 38. (m + 8) + 4
39. 15 + (12 + a)
40. (17 + p) + 9
41. 21 + (k + 16)
42. 6 · (y · 2)
43. 7 · (d · 4)
44. (6 · c) · 8
45. (3 · w) · 5
46. 25s(3)
47. FOOD In food preparation, chefs marinate meat before they cook it because meat absorbs the marinade during the cooking process. Is marinating and cooking meat commutative? Explain. 46 Chapter 1 The Tools of Algebra
EXTRA
PRACTIICE
See pages 762, 794. Self-Check Quiz at pre-alg.com
48. BASKETBALL The Denver Nuggets made the following baskets during the 2005–2006 season. Write an expression that shows how many total baskets the team made during the season.
.UGGETS "ASKETS &REE 4HROWS
0OINT &IELD 'OALS
d
0OINT &IELD 'OALS
3OURCE NBACOM
H.O.T. Problems
49. OPEN ENDED Write a numerical sentence that illustrates the Commutative Property of Multiplication. 50. FIND THE ERROR Kimberly and Carlos are using the Associative Properties of Addition and Multiplication to rewrite expressions. Who is correct? Explain your reasoning. Carlos (2 + 7) · 5 = 2 + (7 · 5)
Kimberly (4 + 3) + 6 = 4 + (3 + 6)
51. CHALLENGE The Closure Property states that because the sum or product of two whole numbers is also a whole number, the set of whole numbers is closed under addition and multiplication. Is the set of whole numbers closed under subtraction and division? If not, give counterexamples. 52.
Writing in Math Explain how real-life situations can be commutative. Give an example of a real-life situation that is commutative and one that is not commutative.
53. How can you find 2 · 198 · 5 mentally? A Use the Associative Property.
54. Which property can NOT be used to show that 10 + 6 + 8 = 10 + 8 + 6?
B Use the Commutative Property.
F Associative Property of Addition
C Use the Additive Identity.
G Associative Property of Multiplication
D Use the Multiplicative Identity.
H Commutative Property of Addition J Multiplicative Identity
ALGEBRA Evaluate each expression if a = 6, b = 4, and c = 5. (Lesson 1-3) 55. a + c - b
56. 8a - 3b
57. 4a - (b + c)
58. 10a ÷ c
59. ALGEBRA Find the value of the expression 4 · (8 + 9) + 6. (Lesson 1-2) 60. MUSIC During a spring concert, the jazz band has 15 minutes to perform. If each of the songs they are considering performing is about 4 minutes long, about how many songs can they play? (Lesson 1-1)
PREREQUISITE SKILL Find each product. 61. 48 × 5
62. 8 × 37
63. 16 × 12
64. 25 × 42
65. 106 × 13
Lesson 1-4 Properties
47
CH
APTER
1
Mid-Chapter Quiz Lessons 1-1 through 1-4
1. GEOMETRY Draw the next two figures in the pattern. (Lesson 1-1)
Find the value of each expression. (Lesson 1-2) 5. 5 + 13 × 2 7. 28 ÷ 4 × 2
6. 7 + 8 - 4 8. 7(3 + 10) - 2 · 6
6(15 + 3) 9. 3[6(12 - 3)] - 17 10. _ 6(9 - 6)
2. MULTIPLE CHOICE The table shows the costs of four weekly magazines. Which magazine saves you the most money if you purchase a yearly subscription instead of an equivalent number of single copies? (Lesson 1-1) Magazine
Cost of Yearly Subscription
Cost of Single Copy
A
$129.99
$2.99
B
$99.95
$2.29
C
$200.95
$3.95
D
$160.00
$3.50
Write a numerical expression for each verbal phrase. (Lesson 1-2) 11. fourteen increased by forty-two 12. six less than the product of seven and nine Evaluate each expression if x = 4 and y = 2. (Lesson 1-3) 13. 5y 15. 7x - 3y
14. x + 10y 16. 9y + 4 - x
16 17. _ x
3x 18. _ x+y
A A
C C
B B
D D
19. SPACE Due to gravity, objects weigh three times as much on Earth as they do on Mercury. How much would an object weigh on Earth if it weighs 25 pounds on Mercury? (Lesson 1-3)
3. MULTIPLE CHOICE The distance between the school and the museum is 24 miles. If the bus driver averages 36 miles per hour, about how long would it take to travel from the school to the museum? (Lesson 1-1)
20. MULTIPLE CHOICE A taxi charges $1.25 for the first mile and then $0.75 for each additional mile m. Which expression can be used to find the total cost Grace would pay for a ride in a taxi? (Lesson 1-3)
F 30 min
H 45 min
A 0.75 + 1.25m
G 40 min
J
50 min
B 1.25 + 0.75m C 0.75m
4. WATER PARKS The table shows the price of admission to a water park. Write an expression to find the cost of admission for 3 adults, 4 children under 8, and 2 senior citizens. (Lesson 1-2) Ticket
D 1.25m Simplify each expression. (Lesson 1-4) 21. (7 + a) + 9
22. 8 · (h · 3)
23. 10 + (g + 20)
24. (12 · p) · 6
Price
Adult
$10
Senior Citizen
$6
Children (ages 8–13)
$8
Children under 8
$5
48 Chapter 1 The Tools of Algebra
25. SCIENCE In chemistry, water is used to dilute acid. Since pouring water into acid could cause spattering and burns, it is important to pour the acid into the water. Is combining acid and water commutative? Explain. (Lesson 1-4)
1-5
Variables and Equations
Main Ideas • Identify and solve open sentences. • Translate verbal sentences into equations.
New Vocabulary equation open sentence solution solving the equation
The table shows the -ENS M &REESTYLE top four places of the men’s 3WIM 4IME 3WIMMER 1500-meter freestyle swimming 4IME S "EHIND B final in the 2004 Olympics. (ACKETT !53 n a. How far behind Hackett was *ENSEN 53! each swimmer? $AVIES '"2 b. Write a rule to describe how you found the time behind for 0RILUKOV 253 each swimmer. -ÕÀVi\ >Ì iÃÓää{°V c. Let s represent the swim time and b represent the amount of time behind Hackett. Rewrite your rule using numbers and variables.
Equations and Open Sentences A mathematical sentence that contains an equals sign (=) is called an equation. A few examples are shown. 5 + 9 = 14
2(6) - 3 = 9
x + 7 = 19
2m - 1 = 13
An equation that contains a variable is an open sentence. An open sentence is neither true nor false. When the variable in an open sentence is replaced with a number, you can determine whether the sentence is true or false. x + 7 = 19
x + 7 = 19
Reading Math Math Symbols The symbol ≠ means is not equal to.
11 + 7 19 Replace x with 11.
12 + 7 19 Replace x with 12. 19 = 19 true
18 ≠ 19 false
When x = 12, this sentence is true.
When x = 11, this sentence is false.
A value for the variable that makes an equation true is called a solution. For x + 7 = 19, the solution is 12. The process of finding a solution is called solving the equation.
EXAMPLE
Solve an Equation
Find the solution of 12 - m = 8. Is it 2, 4, or 7? Replace m with each value. Therefore, the solution of 12 - m = 8 is 4.
Value for m
12 - m = 8
True or False?
2
12 - 2 8
false
4
12 - 4 8
true
7
12 - 7 8
false
1. Find the solution of 18 = n + 7. Is it 8, 9, or 11? Lesson 1-5 Variables and Equations
49
Which value of x makes the equation 2x + 1 = 7 true? A 6
B 5
C 4
D 3
Read the Test Item The solution is the value that makes the equation true. Solve the Test Item Test each value. 2x + 1 = 7 Backsolving The strategy of testing each value is called backsolving. You can also use this strategy with complex equations.
Original equation
2x + 1 = 7
Original equation
2(6) + 1 = 7 Replace x with 6.
2(5) + 1 = 7
Replace x with 5.
13 ≠ 7 False 2x + 1 = 7
11 ≠ 7
False
Original equation
2x + 1 = 7
Original equation
2(4) + 1 = 7 Replace x with 4.
2(3) + 1 = 7
Replace x with 3.
9≠7
7 = 7 True
False
Since 3 makes the equation true, the answer is D.
2. Which value is the solution of 5x - 6 = 14? F3
G4
H5
J6
Personal Tutor at pre-alg.com
Translate Verbal Sentences into Equations Just as verbal phrases can be translated into algebraic expressions, verbal sentences can be translated into equations and then solved.
EXAMPLE
Translate Sentences into Equations
The difference of a number and ten is seventeen. Find the number. Words Variable Equation
The difference of a number and ten is seventeen. Let n = the number. n - 10 = 17
n - 10 = 17 Write the equation. 27 - 10 = 17 Solve mentally: What number minus 10 is 17? n = 27 The solution is 27.
3. The sum of a number and nine is twenty-one. Find the number. As with expressions, equations can also have two variables. The value of one variable changes as a change is made to the other variable. The value of one variable depends on the value of the other variable. A good way to see this relationship is with a table. 50 Chapter 1 The Tools of Algebra
APPLE CIDER A bushel of apples will make approximately 3 gallons of apple cider. The table shows the relationship between the number of bushels of apples and the number of gallons of apple cider.
!PPLE "USHELS L
a. Given b, the number of bushels needed, write an equation that can be used to find g, the number of gallons of apple cider. Words Variables Equation Reasonableness To check the equation, substitute values from the data table and verify that the equation works.
is Let = n
the
'ALLONS OF !PPLE #IDER }
. . Let =
=
b. How many bushels are needed to make 54 gallons of cider? g = 3b
Write the equation.
54 = 3b
Replace g with 54.
54 = 3(18) Solve: What number times 3 is 54? 18 = b
4. PART-TIME JOB Raul’s lawn mower runs for 1.5 hours on one gallon of gas. Given g gallons of gas, write an equation to find h, the number of hours the mower can run. Then find the number of gallons used in 6 hours.
Example 1 (p. 49)
Example 2 (p. 50)
Example 3 (p. 50)
Example 4 (p. 51)
ALGEBRA Find the solution of each equation from the list given. 1. h + 15 = 21; 5, 6, 7 3. k - 25 = 12; 36, 37, 38
2. 13 - m = 4; 7, 8, 9 4. 22 + n = 41; 18, 19, 20
48 5. MULTIPLE CHOICE Find the value of k that makes 6 = _ true. k
A6
B7
C8
D 12
ALGEBRA Define a variable. Then write an equation and solve. 6. A number increased by 8 is 23.
7. Twenty-five is 10 less than a number.
TRAVEL For Exercises 8 and 9, use the following information. The Geiger family is driving at an average speed of 55 miles per hour. The table shows the relationship between the distance driven and the time. 8. Given t, the time in hours, write an equation that can be used to find d, the distance driven. 9. How long would it take them to drive 495 miles?
Extra Examples at pre-alg.com
Time t (hours)
Distance d (miles)
1 2 4 5
55 110 220 275
Lesson 1-5 Variables and Equations
51
HOMEWORK
HELP
For See Exercises Examples 10–17 1 35–36 2 18–23 3 24–27 4
ALGEBRA Find the solution of each equation from the list given. 10. c + 12 = 30; 8, 16, 18
11. g + 17 = 28; 9, 11, 13
12. 23 - m = 14; 7, 9, 11
13. 18 - k = 6; 8, 10, 12
14. 14k = 42; 2, 3, 4
15. 75 = 15n; 3, 4, 5
51 16. _ z = 3; 15, 16, 17
60 17. _ p = 4; 15, 16, 17
ALGEBRA Define a variable. Then write an equation and solve. 18. 19. 20. 21. 22. 23.
The sum of 7 and a number is 23. The sum of 9 and a number is 36. A number minus 10 is 27. The difference between a number and 12 is 54. Twenty-four is the product of 8 and a number. A number times 3 is 45.
PLUMBING For Exercises 24 and 25, use the following information. A standard showerhead uses about 6 gallons of water per minute. The table shows the relationship between time and the water used. 24. Given m, the number of minutes, write an equation that can be used to find g, the number of gallons used. 25. How many minutes elapsed if 72 gallons of water were used? Real-World Link In 1990, the total number of indoor movie screens was about 23,000. Today, there are over 35,000 indoor movie screens. Source: National Association of Theatre Owners
CURRENCY For Exercises 26 and 27, use the following information. In a recent year, 1 U.S. dollar could be exchanged for 0.78 euros. The table shows the relationship between U.S. dollars and euros. 26. Given d, the number of U.S. dollars, write an equation that can be used to find c, the number of euros. 27. How many U.S. dollars can you receive for 7.8 euros?
4AKING A 3HOWER
4IME M MINUTES
7ATER 5SED G GALLONS
U.S. Dollars d
Euros c
1
0.78
2
1.56
3
2.34
5
3.90
Source: exchangerate.com
28. MOVIES Mariko purchased three movie tickets for $24. Define a variable. Then write an equation that can be used to find how much Mariko paid for each ticket. What was the cost of each ticket? EXTRA
PRACTICE
See pages 762, 794. Self-Check Quiz at pre-alg.com
29. HEIGHT During the summer, Ana grew from a height of 65 inches to a height of 68 inches. Define a variable. Then write an equation that can be used to find the increase in height. How many inches did Ana grow? 30. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you write and solve an equation.
52 Chapter 1 The Tools of Algebra RNT Productions/CORBIS
H.O.T. Problems
31. OPEN ENDED Write two different equations whose solutions are each 5. 32. CHALLENGE Write three different equations for which there is no solution that is a whole number. 33. SELECT A TECHNIQUE Student Council had a budget of $650 for the winter dance. It had already spent $439. Which of the following technique(s) might Student Council use to determine how much money it has left? Justify your selection(s). Then use the technique(s) to solve the problem. make a model
34.
calculator
paper/pencil
Writing in Math Explain how solving an open sentence is similar to evaluating an expression. How are they different?
35. If the perimeter of the X V pentagon is 54 centimeters, find V the equation that will n V allow you to find the length of the missing side x. A 2x - 15 + 11 + 8 + 9 = 54 B x - (15 + 11 + 8 + 9) = 54 C x(15 + 11 + 8 + 9) = 54
£x V ££ V
36. Mr. Farley is running a race at a speed of 3.5 miles per hour. Which equation can be used to find the number of miles m he can run in h hours? F h = 3.5 + m G m = 3.5h H h = 3.5m J m = 3.5 + h
D 15 + 11 + 8 + 9 + x = 54
Simplify each expression. (Lesson 1-4) 37. 16 + (7 + d)
38. (4 · p) · 6
39. (23 + k) + 34
40. 10 · (z · 9)
BUSINESS For Exercises 41 and 42, use the following information. (Lesson 1-3) Cornet Cable charges $32.50 a month for basic cable television. Each premium channel selected costs an additional $4.95 per month. 41. Write an expression to find the cost of a month of cable service. 42. How much does cable service cost per year if Abby subscribes to 3 premium channels? Evaluate each expression. (Lesson 1-2) 43. 2 + 3 · 5
44. 8 ÷ 2 · 4
45. 10 - 2 · 4
46. (3 · 4) + (9 · 5)
47. What is the next term in 67, 62, 57, 52, 47, …? (Lesson 1-1)
PREREQUISITE SKILL Evaluate each expression for the given value. (Lesson 1-3) 48. 4x; x = 3
49. 3m; m = 6
50. 12p; p 11
51. 19u; u = 5
Lesson 1-5 Variables and Equations
53
1-6
Ordered Pairs and Relations
Main Ideas • Use ordered pairs to locate points.
Elisa and Nhu are playing a game. The player who gets four Xs or Os in a row wins. 1st move
• Use tables and graphs to represent relations.
X O O
2nd move Nhu places an O at 2 over and 2 up.
New Vocabulary coordinate system y-axis coordinate plane origin x-axis ordered pair x-coordinate y-coordinate graph relation domain range
Elisa places an X at 1 over and 3 up.
3rd move 4th
move
Elisa places an X at 1 over and 1 up.
X
Nhu places an O at 1 over and 2 up. Starting Position
a. Where should Elisa place an X now? Explain your reasoning. b. Suppose (1, 2) represents 1 over and 2 up. How could you represent 3 over and 2 up? c. How are (5, 1) and (1, 5) different? d. Where is a good place to put the next O? e. Work with a partner to finish the game.
Ordered Pairs In mathematics, a coordinate system is used to locate points. The coordinate system is formed by the intersection of two number lines that meet at right angles at their zero points. The vertical number line is called the y-axis.
The origin is at (0, 0), the point at which the number lines intersect.
8 7 6 5 4 3 2 1 O
The coordinate system is also called the coordinate plane.
y
1 2 3 4 5 6 7 8x
The horizontal number line is called the x-axis.
An ordered pair of numbers is used to locate any point on a coordinate plane. The first number is called the x-coordinate. The second number is called the y-coordinate. The x-coordinate corresponds to a number on the x-axis.
54 Chapter 1 The Tools of Algebra
(3, 2)
The y-coordinate corresponds to a number on the y-axis.
To graph an ordered pair, draw a dot at the point that corresponds to the ordered pair. The coordinates are your directions to locate the point. Coordinate System Unless they are marked otherwise, you can assume that each unit on the xand y-axis represents 1 unit. Axes is the plural of axis.
EXAMPLE
Graph Ordered Pairs
Graph each ordered pair on a coordinate system. a. (4, 1)
y
Step 1
Start at the origin.
Step 2
Since the x-coordinate is 4, move 4 units to the right.
Step 3
Since the y-coordinate is 1, move 1 unit up. Draw a dot.
(4 , 1) x
O
b. (3, 0) Step 1
Start at the origin.
Step 2
The x-coordinate is 3. So, move 3 units to the right.
Step 3
Since the y-coordinate is 0, you will not need to move up. Place the dot on the axis.
y
(3 , 0) x
O
Graph each ordered pair on a coordinate system. 1 1A. (2, 3) 1B. (0, 2) 1C. 3, 1_
2
Sometimes a point on a graph is named by using a letter. To identify its location, you can write the ordered pair that represents the point.
EXAMPLE
Identify Ordered Pairs
Write the ordered pair that names each point. a. M Step 1
Start at the origin.
y
Step 2
Move right on the x-axis to find the x-coordinate of point M, which is 2.
Q
Step 3
M N
Move up the y-axis to find the y-coordinate, which is 5.
The ordered pair for point M is (2, 5).
P O
x
b. P The x-coordinate of P is 7, and the y-coordinate is 0. The ordered pair for point P is (7, 0).
2A. N
2B. Q Personal Tutor at pre-alg.com Lesson 1-6 Ordered Pairs and Relations
55
Relations A set of ordered pairs such as {(1, 2), (2, 4), (3, 0), (4, 5)} is a relation. A relation can also be shown in a table or a graph. The domain of the relation is the set of x-coordinates. The range of the relation is the set of y-coordinates. Ordered Pairs Table Graph y (1, 2) x y (2, 4) 1 2 (3, 0) 2 4 (4, 5) The domain is {1, 2, 3, 4}.
The range is {2, 4, 0, 5}.
EXAMPLE Interactive Lab pre-alg.com
3
4
5
x
O
Relations as Tables and Graphs
Express the relation {(0, 0), (2, 1), (1, 3), (5, 2)} as a table and as a graph. Then determine the domain and range. x
y
2
1
1
3
5
2
The domain is {0, 2, 1, 5}, and the range is {0, 1, 3, 2}.
y
O
x
3. Express the relation {(2, 4), (0, 3), (1, 4), (1, 1)} as a table and as a graph. Then determine the domain and range.
PLANTS Some species of bamboo grow 3 feet in one day.
Real-World Link Bamboo is a type of grass. It can vary in height from 1-foot dwarf plants to 100-foot giant timber plants. Source: American Bamboo Society
x
y
(x, y)
1
3
(1, 3)
2
6
(2, 6)
3
9
(3, 9)
4
12
(4, 12)
Bamboo Growth 14 12 10 8 6 4 2 0
y
1 2 3 4 5 6
x
Days
c. Describe the graph. The points appear to fall in a line.
4. CAPACITY One quart is the same as two pints. Make a table of ordered pairs in which the x-coordinate represents the number of quarts and the y-coordinate represents the number of pints for 1, 2, 3, and 4 quarts. Graph the ordered pairs and then describe the graph.
56 Chapter 1 The Tools of Algebra Michael Boys/CORBIS
b. Graph the ordered pairs.
Growth (ft)
a. Make a table of ordered pairs in which the x-coordinate represents the number of days and the y-coordinate represents the amount of growth for 1, 2, 3, and 4 days.
Extra Examples at pre-alg.com
Example 1 (p. 55)
Example 2 (p. 55)
Example 3 (p. 56)
Graph each ordered pair on a coordinate system. 1. H(5, 3)
2. D(6, 0)
3. W(4, 1)
4. Z(0, 1)
Refer to the coordinate system shown at the right. Write the ordered pair that names each point. 5. Q
6. P
7. S
8. R
y
P Q
S
Express each relation as a table and as a graph. Then determine the domain and range.
R x
O
9. {(2, 5), (0, 2), (5, 5)} 10. {(1, 6), (6, 4), (0, 2), (3, 1)} Example 4 (p. 56)
HOMEWORK
HELP
For See Exercises Examples 13–18 1 19–24 2 25–30 3 31–36 4
ENTERTAINMENT For Exercises 11 and 12, use the following information. It costs $4 to buy a student ticket to the movies. 11. Make a table of ordered pairs in which the x-coordinate represents the number of student tickets and the y-coordinate represents the cost for 2, 4, and 5 tickets. 12. Graph the ordered pairs (number of tickets, cost).
Graph each ordered pair on a coordinate system. 13. A(3, 3)
14. D(1, 8)
15. G(2.5, 7)
16. X(7, 2)
17. P(0, 6)
1 18. N 4 _ , 0
Refer to the coordinate system at the right. Write the ordered pair that names each point. 19. C
20. J
21. N
22. T
23. Y
24. B
Express each relation as a table and as a graph. Then determine the domain and range.
2
y
C
T B J
N Y
x
O
25. {(4, 5), (5, 2), (1, 6)}
26. {(6, 8), (2, 9), (0, 1)}
27. {(7, 0), (3, 2), (4, 4), (5, 1)}
28. {(2, 4), (1, 3), (5, 6), (1, 1)}
29. {(0, 1), (0, 3), (0, 5), (2, 0)}
30. {(4, 3), (3, 4), (1, 2), (2, 1)}
AIR PRESSURE For Exercises 31–33, use the table and the following information.
Height (mi)
Pressure (lb/in2)
0 (sea level)
14.7
1
10.2
31. Write a set of ordered pairs for the data.
2
6.4
32. Graph the data.
3
4.3
33. State the domain and the range of the relation.
4
2.7
5
1.6
The air pressure decreases as the distance from Earth increases. The table shows the air pressure for certain distances.
Lesson 1-6 Ordered Pairs and Relations
57
SCIENCE For Exercises 34–36, use the following information. Elizabeth is conducting a physics experiment. She drops a tennis ball from a height of 100 centimeters and then records the height after each bounce. The results are shown in the table. Bounce Height (cm)
1
2
3
4
100
50
25
13
6
34. Write a set of ordered pairs for the data. 35. Graph the data. 36. How high do you think the ball will bounce on the fifth bounce? Explain. SCIENCE For Exercises 37– 40, use the following information and the information at the left. Water boils at sea level at 100°C. The boiling point of water decreases about 5°C for every mile above sea level. Real-World Link Salt Lake City, Utah, is 4330 feet above sea level. Anderson, South Carolina, is 772 feet above sea level. Source: The World Almanac
37. Make a table that shows the boiling point at sea level and at 1, 2, 3, 4, and 5 miles above sea level. 38. Show the data as a set of ordered pairs. 39. Graph the ordered pairs. 40. At about what temperature does water boil in Anderson, South Carolina? in Salt Lake City, Utah? (Hint: 1 mile = 5280 feet) Graph each ordered pair on a coordinate system. 41. W(0.25, 4)
3 43. Y 2_ , 0
42. X(1, 1.3)
1 _ 44. Z 3_ , 31
4
5
4
45. Where are all of the possible locations for the graph of (x, y) if y = 0? If x = 0? EXTRA
PRACTICE
See pages 762, 794.
Graph each relation on a coordinate system. Then find the coordinates of another point that follows the pattern in the graph. 46.
Self-Check Quiz at pre-alg.com
H.O.T. Problems
x
1
3
5
7
y
2
4
6
8
47.
x
2
4
6
y
10
8
6
4
48. OPEN ENDED Give an example of an ordered pair, and identify the x- and y-coordinate. GEOMETRY For Exercises 49–53, draw a coordinate system. 49. Graph (2, 1), (2, 4), and (5, 1). 50. Connect the points with line segments. Describe the figure formed. 51. Multiply each coordinate in the set of ordered pairs by 2. 52. Graph the new ordered pairs. Connect the points with line segments. What figure is formed? 53. MAKE A CONJECTURE How do the figures compare? Write a sentence explaining the similarities and differences of the figures. 54. CHALLENGE Where are all of the possible locations for the graph of (x, y) if x = - 2? 55.
Writing in Math Use the information about ordered pairs found on pages 54–56 to explain how they are used to graph real-life data. Include an example of a situation where ordered pairs are used to graph data.
58 Chapter 1 The Tools of Algebra age fotostock/SuperStock
7ÀÕÌ /i ÕÌiî
56. Felipe drew a graph that shows his daily workout times for the past five days. Find the range of the relation.
57. What relationship exists between the x- and y-coordinates of each of the data points shown on the graph? y
>Þ 7ÀÕÌÃ Îä
Y
Ón ÓÈ Ó{ ÓÓ Óä ä
£
Ó
Î { >Þ
x
x
O
È X
F The y-coordinate varies, and the x-coordinate is always 4.
A {5, 29} B {1, 2, 3, 4, 5}
G The y-coordinate is 4 more than the x-coordinate.
C {20, 21, 26, 28, 29}
H The sum of the x- and y-coordinate is always 4.
D {1, 20}, {2, 21}, {3, 26}, {4, 28}, {5, 29}
J The x-coordinate varies, and the y-coordinate is always 4.
ALGEBRA Solve each equation. (Lesson 1-5) 58. a + 6 = 17
59. 28 = j - 13
54 61. _ n =6
60. 7t = 42
62. Name the property shown by 4 · 1 = 4. (Lesson 1-4) ALGEBRA Evaluate each expression if a = 5, b = 1, and c = 3. (Lesson 1-3) 65 63. _ a
64. a + bc
65. ca - cb
66. 5a - 6c
Write a numerical expression for each verbal phrase. (Lesson 1-2) 67. fifteen less than twenty-one
68. the product of ten and thirty
69. twelve divided into sixty
70. the total of fourteen and nine
71. MANUFACTURING A wagon manufacturer can produce 8000 wagons a day at peak production. Explain how you can find the maximum number of wagons that can be produced in a year. Then find the total. (Lesson 1-1)
PREREQUISITE SKILL Find each quotient. 72. 74 ÷ 2
73. 96 ÷ 8
74. 102 ÷ 3
75. 112 ÷ 4
76. 80 ÷ 16
77. 91 ÷ 13
78. 132 ÷ 22
79. 153 ÷ 17
Lesson 1-6 Ordered Pairs and Relations
59
EXPLORE
1-7
Algebra Lab
Scatter Plots Sometimes, it is difficult to determine whether a relationship exists between two sets of data by simply looking at them. To determine whether a relationship exists, we can write the data as a set of ordered pairs and then graph the ordered pairs on a coordinate system.
ACTIVITY Collect data to investigate whether a relationship exists between height and arm span. Step 1 Work with a partner. Use a meterstick to measure your partner’s height and the length of your partner’s arm span to the nearest centimeter. Record the data in a table like the one shown. Name
Height (cm)
Arm Span (cm)
y
Step 3 Make a list of ordered pairs in which the x-coordinate represents height and the y-coordinate represents arm span. Step 4 Draw a coordinate plane like the one shown and graph the ordered pairs (height, arm span).
Arm Span (cm)
Step 2 Extend the table. Combine your data with that of your classmates.
O
Height (cm)
x
ANALYZE THE RESULTS 1. Does there appear to be a trend in the data? If so, describe the trend. 2. Using your graph, estimate the arm span of a person whose height is 60 inches. 72 inches. 3. How does a person’s arm span compare to his or her height? 4. MAKE A CONJECTURE Suppose the variable x represents height and the variable y represents arm span. Write an expression for arm span. 5. Collect and graph data to determine whether a relationship exists between height and shoe length. Explain your results. 60 Chapter 1 The Tools of Algebra
1-7
Scatter Plots
Main Ideas
New Vocabulary scatter plot
a. What appears to be the trend in sales of movies on videocassette?
Videocassette Sales Number Sold
• Analyze trends in scatter plots.
Suppose you work at a video store. The number of movies on videocassettes sold in a five-year period is shown in the graph.
200 160 120 80 40 0 ’02
’03
b. Estimate the number of movies on videocassette sold for 2008.
’04 Year
’05
’06
Construct Scatter Plots A scatter plot is a graph that shows the relationship between two sets of data. In a scatter plot, two sets of data are graphed as ordered pairs on a coordinate system.
EXAMPLE
Construct a Scatter Plot
TEST SCORES Make a scatter plot of the average SAT math scores from 1995–2004.
Year
Score
‘95
506
‘96
508
‘97
511
‘98
512
‘99
511
‘00
514
‘01
514
‘02
516
‘03
519
‘04
518
Let the horizontal axis, or x-axis, represent the year. Let the vertical axis, or y-axis, represent the score. Then graph ordered pairs (year, score). ÛiÀ>}i -/ -VÀiÃ] £xqÓää{
-VÀi
• Construct scatter plots.
Source: The College Board
9i>À
1. Make a scatter plot of the average ACT scores from 1995 to 2004. Year
‘95
‘96
‘97
‘98
‘99
‘00
‘01
‘02
‘03
‘04
Score
20.8
20.9
21.0
21.0
21.0
21.0
21.0
20.8
20.8
20.9
Source: The College Board
Lesson 1-7 Scatter Plots
61
Analyze Scatter Plots The following scatter plots show the types of relationships or patterns of two sets of data.
Types of Relationships Scatter Plots Data that appear to go uphill from left to right show a positive relationship. Data that appear to go downhill from left to right show a negative relationship.
Positive Relationship
Negative Relationship
y
O
As x increases, y increases.
EXAMPLE
No Relationship
y
x
O
y
x
x
O
As x increases, y decreases.
No obvious pattern.
Interpret Scatter Plots
CAR VALUE Determine whether a scatter plot of the age of a car and the value of a car might show a positive, negative, or no relationship. Explain your answer.
Value (thousands of dollars)
Car Value
As the age of a car increases, the value of the car decreases. So, a scatter plot of the data would show a negative relationship.
Real-World Link
27 y 24 21 18 15 12 9 6 3 0
x 1 2 3 4 5 6 7 8 9 Age (years)
A car loses 15–20% of its value each year.
2. Determine whether a scatter plot of the birth month and birth weight data might show a positive, negative, or no relationship. Explain your answer.
Birth Weight
Birth Weight (lb)
Source: bankrate.com
10 y 9 8 7 6 5 4 3 2 1 0
x J F M A M J J A S O N D Birth Month
You can also use scatter plots to spot trends, draw conclusions, and make predictions about the data. 62 Chapter 1 The Tools of Algebra Toyota
BIOLOGY A biologist recorded the lengths and weights of some largemouth bass. The table shows the results. Length (in.)
9.2
10.9 12.3 12.0 14.1 15.5 16.4 16.9 17.7 18.4 19.8
Weight (lb)
0.5
0.8
0.9
1.3
1.7
2.2
2.5
3.2
a. Make a scatter plot of the data.
Biologist A biologist uses math to study animal populations and monitor trends of migrating animals. For more information, go to pre-alg.com.
4.1
4.8
Largemouth Bass
Weight (lb)
Let the horizontal axis represent length, and let the vertical axis represent weight. Graph the data.
Real-World Career
3.6
b. Does the scatter plot allow you to draw a conclusion about a relationship between the length and weight of a largemouth bass? Explain. As the length of the bass increases, so does its weight. So, the scatter plot shows a positive relationship.
5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 9 10 11 12 13 14 15 16 17 18 19 20 Length (in.)
c. Predict the weight of a bass that measures 22 inches. By looking at the pattern in the graph, we can predict that the weight of a bass measuring 22 inches would be between 5 and 6 pounds.
KEYBOARDING The table shows keyboarding speeds in words per minute (wpm) of 12 students. Experience (weeks)
4
7
8
1
6
3
5
2
9
6
7
10
Speed (wpm)
38
46
48
20
40
30
38
22
52
44
42
55
3A. Make a scatter plot of the data. 3B. Draw a conclusion about the type of relationship the data shows. 3C. Predict the keyboarding speed of a student with 12 weeks of experience. Personal Tutor at pre-alg.com
Example 1 (p. 61)
Example 2 (p. 62)
1. HEALTH CARE The table shows the number of physicians and hospital beds for nine rural counties. Make a scatter plot of the data. Physicians
11
26
10
19
22
9
15
7
1
Hospital Beds
85
67
32
69
49
43
90
49
18
2. Determine whether a scatter plot of hours worked and weekly earnings of a person on the wait staff of a restaurant would show a positive, negative, or no relationship. Explain your answer.
Extra Examples at pre-alg.com David Hiser/Stone/Getty Images
Lesson 1-7 Scatter Plots
63
COMMUNICATION The table shows the number of people in a family and the number of telephone calls made per week.
Example 3 (p. 63)
Number in Family
5
1
4
2
4
6
3
4
7
3
5
8
2
Number of Calls
31
8
26
9
18
34
13
10
25
15
20
36
15
3. Make a scatter plot of the data. 4. Does the scatter plot show a relationship between the number of people in a family and the number of telephone calls made per week? Explain. 5. If a relationship exists, predict the number of calls made during the week for a family of 10.
HOMEWORK
HELP
For See Exercises Examples 6–7 1 8–13 2 14–16 3
6. MUSIC The table shows the number of songs and the total number of minutes on different CDs. Make a scatter plot of the data. Number of Songs
15
18
20
13
12
15
16
17
14
18
20
19
11
14
Total Minutes
64
78
63
70
59
61
77
75
72
71
78
75
63
69
7. OLYMPICS The table shows the winning times for the women’s Olympic 100-meter run. Make a scatter plot of the data. Year
‘28
‘32
‘36
‘48
‘52
‘56
‘60
‘64
‘68
Winning Times (s)
12.2
11.9
11.5
11.9
11.5
11.5
11.0
11.4
11.08
Year
‘72
‘76
‘80
‘84
‘88
‘92
‘96
‘00
‘04
11.07
11.08
11.06
10.97
10.54
10.82
10.94
10.75
10.93
Winning Times (s) Source: olympic.org
Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. 8. size of household and amount of water bill 9. hair color and height 10. temperature and heating costs 11. speed and distance traveled Determine whether the scatter plot of the data shows a positive, negative, or no relationship. Explain your answer. 12.
13.
Ó{ Óä £È £Ó n { ä
64 Chapter 1 The Tools of Algebra
>`i ÕÀ /i Ç
Y
X n £Ó £È Óä Ó{ Ón ÎÓ
ÃÌ v f®
i} Ì v >`i °®
ÕLiÀ v -}Ã
ÕÃV ÓÈ
Y
È x { Î Ó £ ä
X £ä Óä Îä {ä xä Èä Çä ÕÌià ÕÀi`
BASKETBALL For Exercises 14–16, use the information and table below. The number of minutes played and the number of field goal attempts for certain players of the Los Angeles Sparks for the 2004 season are shown. Player
Leslie Mabika Teasley Dixon Whitmore
Minutes Played
Field Goal Attempts
Player
Minutes Played
Field Goal Attempts
1150 965 1105 913 595
451 383 278 269 173
Milton-Jones Thomas Macchi Hodges Masciadri
604 547 410 245 116
161 143 106 52 25
Source: wnba.com
14. Make a scatter plot of the data. 15. Does the scatter plot allow you to draw a conclusion about the relationship between minutes played and field goal attempts? Explain. 16. Suppose a player played 1500 minutes. If a relationship exists, predict the number of field goal attempts for that player. 17. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you draw a scatter plot.
EXTRA
PRACTICE
See pages 763, 794. Self-Check Quiz at pre-alg.com
H.O.T. Problems
The high and low temperatures for your vacation destinations can be shown in a scatter plot. Visit pre-alg.com to continue work on your project.
19. What appears to be the trend in the number of hatchlings between 1965 and 1972? 20. What appears to be the trend between 1972 and 1985?
Average Number of Bald Eagles per Breeding Area 1.4 Number of Eagles
ANIMALS For Exercises 18–21, use the scatter plot shown. 18. Do the data show a positive, negative, or no relationship between the year and the number of bald eagle hatchlings?
1.2 1.0 0.8 0.6 0.4 0 ’65
’70
’75
’80
’85
Year
Source: CHANCE 21. What factors could contribute to the trends displayed in the scatter plot? Predict the number of eagles in years after the data points.
22. OPEN ENDED Draw a scatter plot with ten ordered pairs that shows a negative relationship. 23. CHALLENGE Refer to Example 1 on page 61. Do you think the upward trend in the test scores will continue indefinitely? Why or why not? Explain. NUMBER SENSE What type of relationship is shown on a graph that shows the following values? 24. As x increases, y decreases. 25. As x decreases, y decreases. 26. As x decreases, y increases. 27.
Writing in Math
Explain how you can use scatter plots to help you spot trends. Include real-life examples to illustrate each type of scatter plot. Lesson 1-7 Scatter Plots
65
The scatter plot shows the study time and test scores for the students in Ms. Flores’ math class.
29. Which statement best describes the relationship in the scatter plot? F The longer students studied, the better they did on the test.
Test Score
Study Time and Test Scores
G The shorter students studied, the better they did on the test.
100 95 90 85 80 75 70 65 60 0
H The longer students studied, the worse they did on the test. J There is no relationship between study time and test scores. 10
30
50
70
90 110 20 40 60 80 100 120 Study Time (min)
30. Based on the results, which of the following is a reasonable amount of study time for a student who scores a 75 on the test?
28. Based on the results, which of the following is a reasonable score for a student who studies for 1 hour? A 68
C 87
B 72
D 98
A 10 min
C 61 min
B 32 min
D 88 min
Graph each ordered pair on a coordinate system. (Lesson 1-6)
31. M(3, 2)
32. X(5, 0)
33. K(0, 2)
34. Determine the domain and range of the relation {(0, 9), (4, 8), (2, 3), (6, 1)}. (Lesson 1-6) ANIMALS One year of a dog’s life is equivalent to 7 years of human life, as shown in the table. (Lesson 1-5) 35. Given d, a dog’s age, write an equation to find h, the equivalent human age. 36. What is the age of a dog, if the equivalent human age is 42?
! $OGS ,IFE %QUIVALENT $OGS !GE D (UMAN !GE H
37. ALGEBRA Simplify 15 + (b + 3). (Lesson 1-4) ALGEBRA Evaluate each expression if m = 8 and y = 6. (Lesson 1-3) 38. (2m + 3y) - m
39. 3m + (y - 2) + 3
40. 16 + (mn - 12)
41. COMMUNICATION A telephone tree is set up so that every person calls three other people. Jeffrey needs to tell his co-workers about a time change for a meeting. Suppose it takes 2 minutes to call 3 people. In 10 minutes, how many people will know about the change of time? (Lesson 1-1) 66 Chapter 1 The Tools of Algebra
Graphing Calculator Lab
EXTEND
1-7
Scatter Plots You have learned that graphing ordered pairs as a scatter plot on a coordinate plane is one way to make it easier to “see” if there is a relationship. You can use a TI-83/84 Plus graphing calculator to create scatter plots.
ACTIVITY SCIENCE A zoologist studied extinction times (in years) of island birds. The zoologist wanted to see if there was a relationship between the average number of nests and the time needed for each bird to become extinct on the islands. Use the table of data below to make a scatter plot. Bird Name
Bird Size
Average Number of Nests
Extinction Time
Buzzard
Large
2.0
5.5
Quail
Large
1.0
1.5
Curlew
Large
2.8
3.1
Cuckoo
Large
1.4
2.5
Magpie
Large
4.5
10.0
Swallow
Small
3.8
2.6
Robin
Small
3.3
4.0
Stonechat
Small
3.6
2.4
Blackbird
Small
4.7
3.3
Tree-sparrow
Large
2.0
5.5
Step 1 Enter the data.
The first data pair is (2, 5.5).
• Clear any existing list. KEYSTROKES:
STAT
CLEAR
• Enter the average number of nests as L1 and extinction times as L2. KEYSTROKES:
2
STAT
2.2
5.5
1
…
1.5
… 1.9 Step 2 Format the graph. • Turn on the statistical plot. KEYSTROKES:
2nd [STAT PLOT]
• Select the scatter plot, L1 as the Xlist and L2 as the Ylist. KEYSTROKES:
2nd [L1]
Other Calculator Keystrokes at pre-alg.com Jim Zipp/Photo Researchers
(continued on the next page) Extend 1-7 Graphing Calculator Lab: Scatter Plots
67
Step 3 Graph the data. • Display the scatter plot. KEYSTROKES:
ZOOM 9
• Use the TRACE feature and the left and right arrow keys to move from one point to another.
ANALYZE THE RESULTS RACE . Use the left and right arrow keys to move from 1. Press TRACE one point to another. What do the coordinates of each data point represent?
2. Describe the scatter plot. 3. Is there a relationship between the average number of nests and extinction times? If so, write a sentence or two that describes the relationship. 4. Are there any differences between the extinction times of large birds versus small birds? 5. Separate the data by bird size. Enter average number of nests and extinction times for large birds as lists L1 and L2 and for small birds as lists L3 and L4. Use the graphing calculator to make two scatter plots with different marks for large and small birds. Does your scatter plot agree with your answer in Exercise 4? Explain. For Exercises 6–8, make a scatter plot for each set of data and describe the relationship, if any, between the x- and y-values. 6.
8.
x 70 80 40 50 30 80 60 60 50 40
y 323 342 244 221 121 399 230 200 215 170
7.
x 8 5 9 10 3 4 10 7 6 7
y 89 32 30 18 26 72 51 34 82 60
x
5.2
5.8
6.3
6.7
7.4
7.6
8.4
8.5
9.1
y
12.1
11.9
11.5
9.8
10.2
9.6
8.8
9.1
8.5
9. RESEARCH Find two sets of data on your own. Then determine whether a relationship exists between the data. 68 Chapter 1 The Tools of Algebra
[0, 5] scl:1 by [0, 12] scl:1
CH
APTER
1
Study Guide and Review
wnload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
%XPLORE
0LAN
#HECK
3OLVE
Key Concepts
• Check
(Lesson 1-1)
↓
• Explore ↑
• Plan ↓
↑
Problem-Solving Plan
• Solve
Order of Operations
(Lesson 1-2)
• Step 1 Evaluate the expressions inside grouping symbols. • Step 2 Multiply and/or divide in order from left to right. • Step 3 Add and/or subtract in order from left to right.
Properties
(Lessons 1-4 and 1-5)
For any numbers a, b, and c, the following are true. • a+b=b+a • a·b=b·a • (a + b) + c = a + (b + c) • (a · b) · c = a · (b · c) • a+0=0+a=a • a·1=1·a=a • a·0=0·a=0
Coordinate Plane
(Lesson 1-6)
• x- and y-coordinates are used to indicate a point’s position in a coordinate system. • The domain of a relation is the set of x-coordinates and the range of a relation is the set of y-coordinates.
algebra (p. 37) algebraic expression (p. 37) conjecture (p. 27) coordinate plane (p. 54) counterexample (p. 45) deductive reasoning (p. 45) defining a variable (p. 38) domain (p. 56) equation (p. 49) evaluate (p. 32) inductive reasoning (p. 27) numerical expression (p. 32)
open sentence (p. 49) ordered pair (p. 54)
order of operations (p. 32) origin (p. 54) properties (p. 43) range (p. 56) relation (p. 56) scatter plot (p. 61) simplify (p. 45) solution (p. 49) solving the equation (p. 49) variable (p. 37) x-axis (p. 54) x-coordinate (p. 54) y-axis (p. 54) y-coordinate (p. 54)
Vocabulary Check State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. m + 3n - 4 is an example of a numerical expression. 2. To find the value of a numerical expression, you evaluate that expression. 3. The set of all y-coordinates of a relation is called the domain. 4. 20 + 12 ÷ 4 - 1 × 12 is an example of a numerical expression. 5. The set of all x-coordinates of a relation is called the domain. 6. An ordered pair names a point on a coordinate plane. 7. A counterexample is an example that shows an equation is not true. 8. A relation is a graph that shows the relationship between two sets of data.
Vocabulary Review at pre-alg.com
Chapter 1 Study Guide and Review
69
CH
A PT ER
1
Study Guide and Review
Lesson-by-Lesson Review 1–1
Using a Problem-Solving Plan
(pp. 26–30)
9. FOOD The table below shows the cost of various-sized hams. How much will it cost to buy a ham that weighs 7 pounds? Weight (lb) Cost
1
2
3
4
5
$4.38 $8.76 $13.14 $17.52 $21.90
10. MONEY The cash register drawer has $267 in bills. None of the bills is greater than $10. The drawer has eleven $10 bills and seven fewer $5 bills than $1 bills. How many $5 and $1 bills are in the drawer? Find the next term in each list. 11. 2, 4, 6, 8, 10, …
Example 1 A pay phone at the mall requires 40 cents for a local call. It takes quarters, dimes, and nickels and does not give change. How many combinations of coins could be used to make exact change for a local call? Use the four-step plan to solve the problem. Four-Step Problem-Solving Plan Explore We need to find the number of combinations of quarters, dimes, and nickels that make 40 cents. Plan Make a table showing the different combinations of coins. Solve
12. 5, 8, 11, 14, 17, … 13. 2, 6, 18, 54, 162, … 14. 1, 2, 4, 7, 11, 16, ….
Check
1–2
Numbers and Expressions
2(17 + 4)
18. 3
19. 4[9 + (1 · 16) - 8] 20. 18 ÷ (7 - 4) + 6 21. PROFITS Fuyu, Collin, and Sydney spent $284 to buy supplies to make bracelets. They sold the bracelets for $674. If they split the profits evenly, how much did each person earn?
70 Chapter 1 The Tools of Algebra
Dimes 1 0 4 3 2 1 0
Nickels 1 3 0 2 4 6 8
All of the combinations equal 40 cents and there are no other combinations possible.
(pp. 32–36)
Find the value of each expression. 15. 7 + 3 · 5 16. 36 ÷ 9 - 3 17. 5 · (7 - 2) - 9
Quarters 1 1 0 0 0 0 0
Example 2 Find the value of 3[(10 - 7) + 2]. 3[(10 - 7) + 2] = 3(3 + 2)
Evaluate (10 - 7).
= 3(5)
Add 3 and 2.
= 15
Multiply 3 and 5.
Mixed Problem Solving
For mixed problem-solving practice, see page 794.
1–3
Variables and Expressions
(pp. 37–41)
Evaluate each expression if x = 3, y = 8, and z = 5. 22. y + 6 23. 17 - 2x 6y 24. _ x +9
25. 6x - 2z + 7
26. Translate the phrase nine less than a number into an algebraic expression. 27. PHYSICAL EDUCATION Arturo’s time for climbing the rope was 5 seconds more than half of Brandon’s time. Define the variables and represent this situation as an algebraic expression.
1–4
Properties
5a + 2 = 5(7) + 2
Replace a with 7.
= 35 + 2
Multiply 5 and 7.
= 37
Add 35 and 2.
Example 4 Translate 8 more than 3 divided by a number into an algebraic expression. Words 8 more than 3 divided by a number Variable Let n represent the number. Expression
3 8+_ n
(pp. 43–47)
Name the property shown by each statement. 28. 1 + 9 = 9 + 1 29. 6 + 0 = 6 30. 15 × 0 = 0
31. (x · 8) · 2 = x · (8 · 2)
ALGEBRA Simplify each expression. 32. 3 + (b + 4) 33. 8 · (9 · d) 34. COLLECTIONS Gloria has 58 dolls. If she does not add any dolls to her collection, write a sentence that represents the situation. Then name the property that is illustrated.
1–5
Example 3 Evaluate 5a + 2 if a = 7.
Variables and Equations
Example 5 Name the property shown by the statement. a. (2 + 3) + 6 = 2 + (3 + 6) Associative Property of Addition b. 1 · 6 · 9 = 6 · 1 · 9 Commutative Property of Multiplication Example 6 Simplify (k + 7) + 9. (k + 7) + 9 = k + (7 + 9) = k + 16
Assoc. (+) Add 7 and 9.
(pp. 49–53)
ALGEBRA Solve each equation mentally. 35. n + 3 = 13 36. 24 = 7 + g
Example 7 Find the solution of 26 = 33 - w. Is it 5, 6, or 7?
37. 38 = g + 16
38. 6x = 48
Replace w with each value.
39. 54 = 9h
56 40. _ a = 14
41. BICYCLES A bicycle wheel travels 72 inches in one revolution. Given r, the number of revolutions of the wheel, write an equation to find d, the distance traveled.
Value for w
26 = 33 - w True or False?
5
26 33 - 5
false
6
26 33 - 6
false
7
26 33 - 7
true
Therefore, the solution of 26 = 33 - w is 7.
Chapter 1 Study Guide and Review
71
CH
A PT ER
1 1–6
Study Guide and Review
Ordered Pairs and Relations
(pp. 54–59)
Express each relation as a table and as a graph. Then determine the domain and range. 42. {(2, 3), (6, 1), (7, 5)}
Example 8 Express the relation {(1, 4), (3, 2), (4, 3), (0, 5)} as a table and as a graph. Then determine the domain and range.
43. {(0, 2), (1, 7), (5, 2), (6, 5)}
x
y
44. FAIRS It costs $2 per person to ride the Ferris wheel. Graph the ordered pairs in which the x-coordinate represents the number of people and the y-coordinate represents the cost for 1, 2, and 4 people to ride the Ferris wheel.
1
4
3
2
4
3
5
y
x
O
The domain is {1, 3, 4, 0}, and the range is {4, 2, 3, 5}.
Scatter Plots
(pp. 61–66)
SLEEP The table shows the amount of sleep students received the night before a standardized test and their score on the test. Number of Sleep Hours
8
9.5
8
5
9
7
Score
89
91
94
68
81
77
45. Make a scatter plot of the data. 46. Does the scatter plot allow you to draw a conclusion between sleep time and test score? Explain your reasoning. 47. Predict the score of a person who got 4 hours of sleep the night before the test.
Example 9 TREES The scatter plot shows the approximate heights and circumferences of various giant sequoia trees. Height and Circumference of Giant Sequoia Trees Circumference (ft)
1–7
105 100 95 90 85 80 75 0
220 240 260 280 300 Height (ft)
a. Does the scatter plot allow you to draw a conclusion about the heights and circumferences of Giant Sequoia trees? Explain your reasoning. Yes. As the heights of the trees increase, so do their circumferences. b. If a relationship exists, predict the circumference of a 245-foot Giant Sequoia. The circumference is about 93 ft.
72 Chapter 1 The Tools of Algebra
A PT ER
1
Practice Test
1. STRAWBERRIES Five people can pick 10 baskets of strawberries in one hour. How many baskets of strawberries can 20 people pick in one-half hour? 2. MONEY Mrs. Adams rents a car for a week and pays $79 for the first day and $49 for each additional day. Mr. Lowe rents a car for $350 a week. Which was the better price for a seven-day rental? Explain. ALGEBRA Evaluate each expression if a = 7, b = 3, and c = 5. 3. 42 ÷ [a(c - b)] 4. 5c + (a + 2b) - 8 Name the property shown by each statement. 5. (5 · 6) · 8 = 5 · (6 · 8) 6. x + y = y + x 7. 20 · 1 = 20 8. MULTIPLE CHOICE The table shows the point values of different scoring plays in football. Which set of scoring plays does not result in 30 points? Use the equation 6t + x + 2c + 3f + 2s = 30 to help you. Scoring Play
Points
touchdown (t)
6
extra point (x)
1
two-point conversion (c)
2
field goal (f )
3
safety (s)
2
9. SHOPPING Jacob paid $12 for 6 loaves of bread at the grocery store. Find an equation that can be used to find how much Jacob paid for each loaf of bread. 10. What is the domain of the function shown in the table? ⫺1
1
4
5
y
3
7
13
15 y
E D
11. C 12. D 13. E
C x
O
Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. 14. outside temperature and air conditioning bill 15. number of siblings and height 16. MULTIPLE CHOICE The scatter plot shows semester grades and school days missed for the students in Mr. Hernandez’s math class. Which of the following is a reasonable score for a student who missed 3 days? Attendance 100
A 2 touchdowns, 1 two-point conversion, 5 field goals
x
Refer to the coordinate system at the right. Write the ordered pair that names each point.
y
95 Semester Score
CH
90 85 80 75 70
B 3 touchdowns, 2 extra points, 2 field goals, 2 safeties
2
4
6 8 10 12 14 16 x Days Missed
C 4 touchdowns, 4 extra points, 1 safety D 3 touchdowns, 2 extra points, 1 twopoint conversion, 2 field goals, 1 safety
Chapter Test at pre-alg.com
F 85 G 83
H 91 J 95
Chapter 1 Practice Test
73
A PT ER
Standardized Test Practice
1
Chapter 1
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The equation c = 0.95t represents c, the cost of t tickets on a subway. Which table can be used to find the values that fit this equation? A Cost of Subway Tickets t
1
2
3
4
c
$1.95
$2.95
$3.95
$4.95
B
3. The manager of an ice cream shop has recorded the average number of snow cones sold per hour based on the outside temperature. The results are shown in the scatter plot. Which description best represents the relationship of the data? -Ü ià -` *iÀ ÕÀ
CH
Cost of Subway Tickets t
1
2
3
4
c
$1.95
$2.90
$3.85
$4.80
C
Cost of Subway Tickets t
1
2
3
4
c
$0.95
$1.95
$2.95
$3.95
D
Cost of Subway Tickets t
1
2
3
4
c
$0.95
$1.90
$2.85
$3.80
2. James, Kyle, and Tommy scored a total of 39 points in their last basketball game. James scored three times as many points as Kyle, and Tommy scored 3 fewer points than James. Which is a reasonable conclusion about the number of points scored by the basketball players? F James scored the most points. G Tommy and Kyle scored the same number of points. H James scored exactly half the total number of points. J Tommy had the fewest points.
Question 2 Eliminate any answers that do not
make sense. For example, in this problem, you can eliminate H because half of 39 points would be 19.5 points and it is impossible to score half a point.
74 Chapter 1 The Tools of Algebra
A B C D
Ü xä
Y
{x {ä Îx Îä Óx ä
X Çä Çx nä nx ä x "ÕÌÃ`i /i«iÀ>ÌÕÀi ®
Negative trend No trend Positive trend Cannot be determined
4. Edward and his sister agreed to split their lunch bill evenly. The subtotal came to $24.70, the sales tax was $1.48, and they left a tip of $5.00. How much did each person owe? F $22.04 G $18.73 H $15.59 J $14.20 5. GRIDDABLE Find the sum of the whole numbers mentally. 28 + 41 + 22 + 19 6. A plumber had a 5-meter length of pipe in his truck. He used 120 centimeters of the pipe on a new hot water heater and 240 centimeters to repair a floor drain. How many meters of the pipe are left? A 0.14 m B 1.4 m C 140 m D 280 m Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
7. The expression below can be used to generate the terms of a pattern where n is the term number. 2n - 4
12. What are the coordinates of point D on the coordinate grid below?
What is the fifth term of the pattern?
A B C D
F 6 G8 H 10
(3, 2) (6, -2) (-2, 6) (-4, -3)
C
654321 1 2 3 E 4 5 6
J 12
y
F B x O 1 2 3 4 5 6
G
A
Pre-AP
8. GRIDDABLE An elevator began on the fifth floor of a hotel. The elevator then traveled 6 floors up, 3 floors down, 8 floors down, 5 floors up, 4 floors down, and then 7 floors up. What was the number of the floor where the elevator finally stopped?
Record your answers on a sheet of paper. Show your work. 13. The table below shows the results of a survey about the average time that individual students spend studying on weeknights.
9. Which property is illustrated by the equation below? 4+6=6+4 A B C D
6 5 4 3 2 1
D
Associative Property Commutative Property Distributive Property Identity Property
10. Suppose your sister has 3 more CDs than you do. Which equation represents the number of CDs that you have? Let y represent your CDs and s represent your sister’s CDs. F y=s+3 G y=3-s H y=s-3 J y = 3s
Grade
Time (min)
Grade
Time (min)
2
20
6
60
2
15
6
45
2
20
6
55
4
30
6
60
4
20
8
70
4
25
8
80
4
40
8
75
4
30
8
60
a. Make a scatter plot of the data. b. What are the coordinates of the point that represents the longest time spent on homework? c. Does a relationship exist between grade level and time spent studying? If so, write a sentence to describe the relationship. If not, explain why not.
11. GRIDDABLE What is the next term in the pattern 4, 12, 36, 108, . . . ?
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
13
Go to Lesson...
1-6
1-1
1-7
1-2
1-4
1-2
1-3
1-1
1-4
1-5
1-1
1-6
1-7
Chapter 1 Standardized Test Practice
75
Integers
2 •
Select and use appropriate operations to solve problems and justify solutions.
•
Communicate mathematics through informal and mathematical language, representations, and models.
Key Vocabulary absolute value (p. 80) integers (p. 78) negative number (p. 78) opposites (p. 88)
Real-World Link Golf The scoring system in golf is based on integers. A positive score is over par, a negative score is under par, and a score of 0 is par.
Operations with Integers Make this Foldable to help you organize your notes about operations with integers. Begin with a sheet of grid paper.
1 Fold in half.
2 Fold the top to the bottom twice.
3 Open and cut along the second fold to make four tabs.
76 Chapter 2 Integers Scott Halleran/Getty Images
4 Fold lengthwise. Draw a number line on the outside. Label each tab as shown.
GET READY for Chapter 2 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Example 1
Evaluate each expression if a = 4, b = 10, and c = 8. (Lesson 1-3) 1. a + b + c
2. bc - ab
3. b + ac
4. 4c + 3b
Evaluate a - 2b + 3c if a = 1, b = 2, and c = 3. a - 2b + 3c = 1 - 2(2) + 3(3) Replace a with 1, b with 2, and c with 3.
5. SALES Laura sold three times as many bottles of water on Sunday as on Saturday. How many bottles did she sell Saturday if she sold 120 bottles Sunday? (Lesson 1-3)
6. 34, 40, 46, 52, 58, …
13. (0, 2)
14. (6, 1)
15. (6, 4)
16. (4, 2)
17. (1, 1)
18. (2, 5)
y
N
V M
U T P L
Q S
11 +5
16 +5
21 +5
? +5
The next term is 21 + 5 or 26.
Example 3
Use the coordinate plane to name each ordered pair. (Prerequisite Skill) 12. (3, 4)
Simplify.
6
8. PHONE The telephone company charges $0.30 for the first minute and $0.15 for each additional minute. How much would it cost to talk for 10 minutes? (Lesson 1-1)
11. (5, 5)
=6
Find the next term in the list. 6, 11, 16, 21, …
7. 135, 120, 105, 90, 75, …
10. (5, 2)
Multiply.
Example 2
Find the next term in each list. (Lesson 1-1)
9. (1, 3)
=1-4+9
R
O
x
ANIMALS A mole can dig a tunnel 300 feet
long in one night. (Prerequisite Skill) 19. Make a table of ordered pairs in which the x-coordinate represents the number of nights and the y-coordinate represents the tunnel length for 1, 2, 3, and 4 nights.
Use the coordinate plane to write the ordered pair that names point A.
Y
Step 1 Start at the origin.
!
Step 2 Move right on " the x-axis to find the x-coordinate of point A, which is 4.
X
Step 3 Move up the y-axis to find the y-coordinate, which is 1. The ordered pair for point A is (4, 1).
20. Graph the ordered pairs. Chapter 2 Get Ready for Chapter 2
77
2-1
Integers and Absolute Value
Main Ideas
• Find the absolute value of an expression.
New Vocabulary negative number integers coordinate inequality absolute value
The western United States was unusually dry in 2002. In the graph, a value of -6 represents 6 inches below the normal rainfall.
Rainfall, 2002 Albuquerque, NM Normal Rainfall
a. What does a value of -3 represent?
Rainfall (in.)
• Compare and order integers.
b. Which city was farthest from its normal rainfall? c. How could you represent 5 inches above normal rainfall?
4 2 0
Denver, CO
Salt Lake City, UT
⫺2 ⫺4 ⫺6 ⫺8
Cities
Sources: weather.com, wonderground.com
Compare and Order Integers With normal rainfall as the starting point of 0, you can express 6 inches below normal as 0 - 6, or -6. A negative number is a number less than zero.
Reading Math Integers Read –6 as negative 6. A positive integer like 6 can be written as +6. It is usually written without the + sign, as 6.
Negative numbers like -6, positive numbers like +6, and zero are members of the set of integers. Integers can be represented as points on a number line. positive integers
negative integers Numbers to the left of zero are less than zero.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
Numbers to the right of zero are greater than zero.
Zero is neither negative nor positive.
This set of integers can be written {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}, where … means continues indefinitely.
EXAMPLE
Write Integers for Real-World Situations
Write an integer for each situation. a. 500 feet below sea level
The integer is -500.
b. a temperature increase of 12°
The integer is +12.
c. a loss of $240
The integer is -240.
1A. a loss of 8 yards 78 Chapter 2 Integers
1B. a deposit of $15
To graph integers, locate the points named by the integers on a number line. The number that corresponds to a point is called the coordinate of that point. graph of a point with coordinate -4
-6
Reading Math Inequality Symbols Read the symbol < as is less than. Read the symbol > as is greater than.
-5
-4
-3 -2
graph of a point with coordinate 2
-1
1
2
3
4
5
6
Notice that the numbers on a number line increase as you move from left to right. This can help you determine which of two numbers is greater. Words
-4 is less than 2.
OR
-4 < 2
OR
Symbols
2 is greater than -4. 2 > -4
The symbol points to the lesser number.
Any mathematical sentence containing < or > is called an inequality. An inequality compares numbers or quantities.
EXAMPLE
Compare Two Integers
Use the integers graphed on the number line below. -6 -5 -4 -3 -2 -1
1
2
3
4
5
6
a. Write two inequalities involving -3 and 4. Since -3 is to the left of 4, write -3 < 4. Since 4 is to the right of -3, write 4 > -3. b. Replace the with or = in -5 -1 to make a true sentence. -1 is greater since it lies to the right of -5. So write -5 < -1.
2A. Write two inequalities involving -2 and -6. 2B. Replace the with < or > in 2 -1 to make a true sentence.
Real-World Link Annika Sorenstam won the 2004 LPGA Championship at 13 under par. She was the LPGA’s leading money winner from 2001 to 2004. Source: LPGA.com
GOLF The top ten fourth-round scores of the 2004 LPGA Championship tournament were 0, +1, -5, -2, -1, +4, +2, +3, +5, and -3. Order the scores from least to greatest. Graph each integer on a number line. -5
-4
-3
-2 -1
1
2
3
4
5
Write the numbers as they appear from left to right. The scores -5, -3, -2, -1, 0, +1, +2, +3, +4, and +5 are in order from least to greatest.
3. GOLF The top ten fourth-round scores of the 2004 PGA Championship were +4, -2, +6, +1, -4, -3, +5, -1, +2, and +3. Order the scores from least to greatest. Personal Tutor at pre-alg.com Lesson 2-1 Integers and Absolute Value Jonathan Daniel/Getty Images
79
Absolute Value On the number line, notice that -5 and 5 are each 5 units from 0, even though they are on opposite sides of 0. Numbers that are the same distance from zero have the same absolute value. 5 units
-6
Common Misconception It is not always true that the absolute value of a number is the opposite of the number. Remember that absolute value is always positive or zero.
-5
-4
-3 -2
5 units
-1
1
2
3
4
5
6
The symbol for absolute value is two vertical bars on either side of the number.
⎪5⎥ = 5 The absolute value of 5 is 5. ⎪-5⎥ = 5 The absolute value of -5 is 5. Absolute Value Words
The absolute value of a number is the distance the number is from zero on the number line. The absolute value of a number is always greater than or equal to zero.
Examples
⎪5⎥ = 5
EXAMPLE
⎪-5⎥ = 5
Expressions with Absolute Value
Evaluate each expression. a. ⎪-8⎥ 8 units
-10
-8
-6
-4
-2
2
⎪-8⎥ = 8 The graph of -8 is 8 units from 0. b. ⎪9⎥ + ⎪-7⎥
The absolute value of 9 is 9.
⎪9⎥ + ⎪-7⎥ = 9 + 7 The absolute value of -7 is 7. = 16
Simplify.
4B. ⎪-4⎥ - ⎪3⎥
4A. ⎪-3⎥
Since variables represent numbers, you can use absolute value notation with algebraic expressions involving variables.
EXAMPLE
Algebraic Expressions with Absolute Value
ALGEBRA Evaluate ⎪x⎥ - 3 if x = -5. ⎪x⎥- 3 = ⎪-5⎥ - 3
Replace x with -5.
=5-3
The absolute value of -5 is 5.
=2
Simplify.
5. Evaluate ⎪y⎥ + 8 if x = -7. 80 Chapter 2 Integers
Extra Examples at pre-alg.com
Example 1 (p. 78)
Example 2 (p. 79)
Write an integer for each situation. Then graph on a number line. 1. 8° below zero
2. a 15-yard gain
Write two inequalities using the numbers in each sentence. Use the symbols < or >. 3. -7° is colder than 3°. Replace each 5. -18
Example 3 (p. 79)
HELP
For See Exercises Examples 16–21 1 22–33 2 34–39 3 40–51 4 52–57 5
{ä
xä
Èä
VViiÀ>Ì Éà ® £È Σ £ä È
Î{
ä
-9
£ä
Óä
Îä
Çä
nä
ä
ää
£ä £Óä «
HOMEWORK
/i î
nä
{ä
(p. 80)
7. 9
-3
9. TEST DRIVES The table shows the recorded acceleration for a new car at regular intervals. Order the accelerations from least to greatest. Îä
Example 5
6. 0
8. Order the integers {28, -6, 0, -2, 5, -52, 115} from least to greatest.
£ä ä
(p. 80)
with , or = to make a true sentence.
-8
Óä
Example 4
4. -6 is greater than -10.
Ó
ÓÎ £ Ó
Evaluate each expression. 11. ⎪10⎥ - ⎪-4⎥
10. ⎪-10⎥
12. ⎪16⎥ + ⎪-5⎥
ALGEBRA Evaluate each expression if a = -8 and b = 5. 13. 9 + ⎪a⎥
14. ⎪a⎥ - b
15. 2⎪a⎥
Write an integer for each situation. Then graph on a number line. 16. a bank withdrawal of $100
17. a loss of 6 pounds
18. a salary increase of $250
19. a gain of 9 yards
20. 12° above zero
21. 5 seconds before liftoff
Write two inequalities using the numbers in each sentence. Use the symbols < or >. 22. 3 meters is taller than 2 meters. 23. A temperature of -5°F is warmer than a temperature of -10°F. 24. 55 miles per hour is slower than 65 miles per hour. 25. A 4-yard loss is less than no gain. Replace each 26. -6
-2
30. -18
8
with , or = to make a true sentence. 27. -10 31. 5
-13
-23
28. 0 32. ⎪9⎥
-9
⎪-9⎥
29. 14
33. ⎪-20⎥
⎪-4⎥
Order the integers in each set from least to greatest. 34. {5, 0, -8}
35. {-15, -1, -2, -4}
36. {19, -16, 4, 62, -80}
37. {41, -14, 50, -23, -20}
38. {24, 5, -46, 9, 0, -3}
39. {98, -57, -60, 38, 188} Lesson 2-1 Integers and Absolute Value
81
Evaluate each expression. 40. ⎪-15⎥
41. ⎪46⎥
42. -⎪20⎥
43. -⎪5⎥
44. ⎪0⎥
45. ⎪7⎥
46. ⎪-5⎥ + ⎪4⎥
47. ⎪0⎥ + ⎪-2⎥
48. ⎪15⎥ - ⎪-1⎥ 49. ⎪0 + 9⎥
50. ⎪9 - 5⎥ - ⎪6 - 8⎥ 51. -⎪-6 + 1⎥ - ⎪5 - 6⎥
ALGEBRA Evaluate each expression if a = 0, b = 3, and c = -4.
Real-World Link The Marianas Trench in the Pacific Ocean is the deepest part of all of the oceans at 35,840 feet.
52. 14 + ⎪b⎥
53. ⎪c⎥ - a
54. a + b + ⎪c⎥
55. ab + ⎪-40⎥
56. ⎪c⎥ - b
57. ⎪ab⎥ + b
58. GEOGRAPHY The Caribbean Sea has an average depth of 8448 feet below sea level. Use an integer to express this depth. ANALYZE TABLES For Exercises 59–62, use the table. Record Lowest Temperatures by State
Source: U.S. Department of Defense
State
Station
Date
Temperature (°F)
Alaska Montana Wisconsin
Prospect Creek Camp Rogers Pass Danbury
Jan. 23, 1971 Jan. 20, 1954 Jan. 24, 1922
-80 -70 -54
59. Graph the temperatures on a number line. 60. Compare the lowest temperature in Alaska and the lowest temperature in Wisconsin using the < symbol. 61. Compare the lowest temperature in Montana and the lowest temperature in Wisconsin using the > symbol. 62. Write the temperatures in order from greatest to least. Graph each set of integers on a number line. 63. {0, -2, 4}
64. {-3, 1, 2, 5}
65. {-2, -4, -5, -8}
66. {-4, 0, 6, -7, -1}
67. Name the coordinates of each point graphed on the number line.
B -6
EXTRA
PRACTICE
See pages 763, 795. Self-Check Quiz at pre-alg.com
H.O.T. Problems
-4
D -2
A 0
C 2
4
6
68. SOLAR SYSTEM The average temperature of Saturn is -218°F while the average temperature of Jupiter is -162°F. Which planet has the lower average temperature? Explain. 69. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would compare and order integers. 70. OPEN ENDED Write two inequalities using integers. 71. NUMBER SENSE Explain how to find the number of units apart -4 and 5 are on a number line. 72. Which One Doesn’t Belong? Identify the expression that does not belong with the other three. Explain your reasoning.
⎪12 – ⎪–4⎥⎥ 82 Chapter 2 Integers NOAA
⎪–2⎥ + ⎪6⎥
–⎪7 + 1⎥
⎪–8⎥
CHALLENGE Consider two numbers A and B on a number line. 73. Is it always, sometimes, or never true that the distance between A and B equals the distance between |A| and |B|? Explain. 74. Assume A > B. Is it always, sometimes, or never true that A - |B| ≤ A + B? Explain. 75.
Writing in Math Use the information about integers on page 78 to explain how they can be used to model real-world situations. Include an explanation of how integers are used to describe rainfall.
76. What is the temperature shown on the thermometer at the right?
5
78. Which of the following describes the absolute value of -2°? F It is the distance from -2 to 2 on the thermometer.
A 8°F
⫺5
B 7°F
⫺10
G It is the distance from -2 to 0 on the thermometer.
C -7°F
H It is the actual temperature outside when the thermometer reads -2°.
D -8°F
J None of these describes the absolute value of -2°.
77. GRIDDABLE How many units apart are -4 and 3 on a number line?
Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. Explain your answer. (Lesson 1-7) 79. height and arm length 80. birth month and weight Express each relation as a table and as a list of ordered pairs. (Lesson 1-6)
81.
82.
y
O
x
y
x
O
Name the property shown by each statement. (Lesson 1-4)
83. 20(18) = 18(20)
84. 9(8)(0) = 0
PREREQUISITE SKILL Find each sum or difference. 86. 18 + 29 + 46 87. 232 + 156 89. 36 - 19 90. 479 - 281
85. 3ab = 3ba
88. 451 + 629 + 1027 91. 2011 - 962
Lesson 2-1 Integers and Absolute Value
83
EXPLORE
2-2
Algebra Lab
Adding Integers In a set of algebra tiles, £ represents the integer 1, and ⫺1 represents the integer -1. You can use algebra tiles and an integer mat to model operations with integers.
ACTIVITY 1 The following example shows how to find the sum -3 + (-2) using algebra tiles. Remember that addition means combining. The expression -3 + (-2) tells you to combine a set of 3 negative tiles with a set of 2 negative tiles. #OMBINE THE TILES ON THE MAT 3INCE THERE ARE NEGATIVE TILES ON THE MAT THE SUM IS
0LACE NEGATIVE TILES AND NEGATIVE TILES ON THE MAT
•
Therefore, -3 + (-2) = -5. There are two important properties to keep in mind when you model operations with integers. • When one positive tile is paired with one negative tile, the result is called a zero pair. • You can add or remove zero pairs from a mat because removing or adding zero does not change the value of the tiles on the mat. The following example shows how to find the sum -4 + 3. 2EMOVE THE ZERO PAIRS
0LACE NEGATIVE TILES AND POSITIVE TILES ON THE MAT
£
£
£
£
£
£
£
£ {
Î
Therefore, -4 + 3 = -1. 84 Chapter 2 Integers
3INCE THERE IS ONE NEGATIVE TILE REMAINING THE SUM IS
£
£
£
£
£
£
£ { Î
{ Î £
EXERCISES Use algebra tiles to model and find each sum. 1. -2 + (-4)
2. -3 + (-5)
3. -6 + (-1)
4. -4 + (-5)
5. -4 + 2
6. 2 + (-5)
7. -1 + 6
8. 4 + (-4)
ACTIVITY 2 The Addition Table was completed using algebra tiles. In the highlighted portions of the table, the addends are -3 and 1, and the sum is -2. So, -3 + 1 = -2. You can use the patterns in the Addition Table to learn more about integers. Addition Table
+
4
3
2
1
-1
-2
-3
addends
-4
⎫
4
8
7
6
5
4
3
2
1
3
7
6
5
4
3
2
1
-1
2
6
5
4
3
2
1
-1
-2
1
5
4
3
2
1
-1
-2
-3
4
3
2
1
-1
-2
-3
-4
-1
3
2
1
-1
-2
-3
-4
-5
-2
2
1
-1
-2
-3
-4
-5
-6
-3
1
-1
-2
-3
-4
-5
-6
-7
-4
-1
-2
-3
-4
-5
-6
-7
-8
⎭
⎬ sums
addends
ANALYZE THE RESULTS 9. MAKE A CONJECTURE Locate all of the positive sums in the table. Describe the addends that result in a positive sum. 10. MAKE A CONJECTURE Locate all of the negative sums in the table. Describe the addends that result in a negative sum. 11. MAKE A CONJECTURE Locate all of the sums that are zero. Describe the addends that result in a sum of zero. 12. The Identity Property says that when zero is added to any number, the sum is the number. Does it appear that this property is true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. 13. The Commutative Property says that the order in which numbers are added does not change the sum. Does it appear that this property is true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. 14. The Associative Property says that the way numbers are grouped when added does not change the sum. Is this property true for addition of integers? If so, write two examples that illustrate the property. If not, give a counterexample. Explore 2-2 Algebra Lab: Adding Integers
85
2-2
Adding Integers
Main Ideas • Add two integers. • Add more than two integers.
In football, forward progress is represented by a positive integer. Being pushed back is represented by a negative integer. On the first play a team loses 5 yards and on the second play they lose 2 yards. ⫺2
New Vocabulary opposites additive inverse
⫺5
⫺9 ⫺8⫺7⫺6⫺5 ⫺4⫺3 ⫺2 ⫺1 0 1 2
40
50
a. What integer represents the total yardage on the two plays? b. Write an addition sentence that describes this situation.
Add Integers The equation -5 + (-2) = -7 is an example of adding two integers with the same sign. Notice that the sign of the sum is the same as the sign of the addends. Recall that the numbers you add are called addends. The result is called the sum.
EXAMPLE
Add Integers on a Number Line
Find -2 + (-3) ⫺3
⫺2
-7 -6-5-4 -3 -2 -1 0 1 2
Start at zero. Move 2 units to the left. From there, move 3 more units to the left.
-2 + (-3) = -5
1A. -3 + (-4)
1B. -6 + (-14)
This example suggests a rule for adding integers with the same sign. Adding Integers with the Same Sign
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
86 Chapter 2 Integers
Words
To add integers with the same sign, add their absolute values. The sum is: • positive if both integers are positive. • negative if both integers are negative.
Examples -5 + (-2) = -7
6+3=9
EXAMPLE
Add Integers with the Same Sign
Find -4 + (-5). -4 + (-5) = -9 Add ⎪-4⎥ and ⎪-5⎥. The sum is negative. Find each sum. 2B. -1 + (-12)
2A. -8 + (-2)
A number line can also help you understand how to add integers with different signs.
EXAMPLE
Add Integers on a Number Line
Find each sum. b. 2 + (-3)
a. 7 + (-4) Adding Integers on a Number Line Always start at zero. Move right to model a positive integer. Move left to model a negative integer.
⫺4
⫺3
7 -2 -1 0 1 2 3 4 5 6 7
2 -4 -3 -2-1 0 1 2 3 4 5
Start at zero. Move 7 units to the right. From there, move 4 units to the left.
Start at zero. Move 2 units to the right. From there, move 3 units to the left.
7 + (-4) = 3
2 + (-3) = -1
3A. 5 + (-2)
3B. 4 + (-8)
Personal Tutor at pre-alg.com
Notice how the sums in Example 3 relate to the addends. The sign of the sum is the same as the sign of the addend with the greater absolute value. Adding Integers with Different Signs To add integers with different signs, subtract their absolute values. The sum is:
• positive if the positive integer’s absolute value is greater. • negative if the negative integer’s absolute value is greater.
EXAMPLE
Add Integers with Different Signs
Find each sum. a. -8 + 3
b. 10 + (-4)
-8 + 3 = -5
10 + (-4) = 6
To find -8 + 3, subtract ⎪3⎥ from ⎪-8⎥. The sum is negative because ⎪-8⎥ > ⎪3⎥.
To find 10 + ⎪-4⎥, subtract ⎪-4⎥ from ⎪10⎥. The sum is positive because ⎪10⎥ > ⎪-4⎥.
4A. -9 + 4 Extra Examples at pre-alg.com
4B. 12 + (-5) Lesson 2-2 Adding Integers
87
ASTRONOMY During the night, the average temperature on the moon is -140°C. By noon, the average temperature has risen 252°C. What is the average temperature on the moon at noon? Temperature at night
Words Variable
plus
increase by noon
equals
252
=
temperature at noon
Let x = the temperature at noon.
Equation
-140
+
x
Solve the equation. Estimate -140 + 250 = 110.
Real-World Link The temperatures on the moon are so extreme because the moon does not have any atmosphere to trap heat.
-140 + 252 = x To find the sum, subtract ⎪-140⎥ from 252. 112 = x The sum is positive because ⎪252⎥ > ⎪-140⎥. The average temperature at noon is 112°C. The solution is reasonable to the estimate.
5. SUBMARINES A submarine was at a depth of 103 feet below the surface of the water. It rose 68 feet. What is its current depth?
Add More Than Two Integers Two numbers with the same absolute value but different signs are called opposites. For example, -4 and 4 are opposites. An integer and its opposite are also called additive inverses. Additive Inverse Property Words
The sum of any number and its additive inverse is zero.
Symbols
x + (-x) = 0
EXAMPLE Adding Mentally One way to add mentally is to group the positive addends together and the negative addends together. Then add to find the sum. Also look for addends that are opposites. You can always add in order from left to right.
NASA
6 + (-6) = 0
Add Three or More Integers
Find each sum. a. 9 + (-3) + (-9) 9 + (-3) + (-9) = 9 + (-9) + (-3) Commutative Property = 0 + (-3) Additive Inverse Property: 9 + (-9) = 0 = -3 Identity Property of Addition b. -4 + 6 + (-3) + 9 Commutative Property -4 + 6 + (-3) + 9 = -4 + (-3) + 6 + 9 = [-4 + (-3)] + (6 + 9) Associative Property = -7 + 15 or 8 Simplify.
6A. 4 + (-2) + (-7) 88 Chapter 2 Integers
Example
6B. -10 + 3 + (-7) + 12
Examples 1– 4 (pp. 86–87)
Example 5 (p. 88)
Example 6 (p. 88)
HOMEWORK
HELP
For See Exercises Examples 13–22 1, 2 23–30 3, 4 31, 32 5 33–40 6
Find each sum. 1. -2 + (-4)
2. -10 + (-5)
3. -14 + (-4)
4. 7 + (-2)
5. 11 + (-3)
6. 8 + (-5)
7. 2 + (-16)
8. 9 + (-12)
9. -15 + 4
10. FOOTBALL A team gained 4 yards on one play. On the next play, they lost 5 yards. Write an addition sentence to find the total yardage. Find each sum. 11. 8 + (-6) + 2
12. -6 + 5 + (-10)
Find each sum. 13. -4 + (-1)
14. -5 + (-2)
15. -4 + (-6)
16. -3 + (-8)
1 7. -7 + (-8)
18. -12 + (-4)
19. -9 + (-14)
20. -15 + (-6)
21. -11 + (-15)
22. -23 + (-43)
23. 8 + (-5)
24. 6 + (-4)
25. 3 + (-7)
26. 4 + (-6)
27. -15 + 6
28. -5 + 11
29. 18 + (-32)
30. -45 + 19
Write an addition sentence for each situation. Then find the sum. 31. GAME SHOWS A contestant has -1500 points. Suppose he loses another 1250 points. 32. STOCKS A stock price increases $6. It then decreases $10. Find each sum. 33. 6 + (-9) + 9
34. 7 + (-13) + 4
35. -9 + 16 + (-10)
36. -12 + 18 + (-12)
37. 14 + (-9) + 6
38. 28 + (-35) + 4
39. -41 + 25 + (-10)
40. -18 + 35 + (-17)
41. MONEY The starting balance in a checking account was $50. What was the balance after checks were written for $25 and for $32? Use estimation to determine whether your answer is reasonable. 42. HIKING Sally starts hiking at an elevation of 324 feet. She descends to an elevation of 201 feet and then ascends to an elevation 55 feet higher than where she began. She descends 183 feet. Describe the overall change in elevation. Find each sum. 43. 18 + (-13)
44. -27 + 19
45. -25 + (-12) Lesson 2-2 Adding Integers
89
EXTRA
PRACTIICE
See pages 763, 795.
POPULATION For Exercises 46 and 47, use the table that shows the change in population of several cities from 2002 to 2003. #ITY %L 0ASO 48 3AN *OSE #! ,EXINGTON +9 #OLUMBIA 3#
Self-Check Quiz at pre-alg.com
0OPULATION
#HANGE AS OF
-ÕÀVi\ 4HE 7ORLD !LMANAC
46. What was the population in each city in 2003? 47. What was the total change in population of these cities?
H.O.T. Problems
48. OPEN ENDED Give an example of two integers that are additive inverses. 49. CHALLENGE True or false? ⫺n always names a negative number. If false, give a counterexample. CHALLENGE Name the property illustrated by each of the following. 50. a(b + (-b)) = (b + (-b))a 52.
Writing in Math
51. a(b + (-b)) = 0
Explain how a number line can help you add integers.
53. A Guadelupe bass was swimming underwater at a depth of 12 feet. It rose 3 feet, dropped 5 feet, rose 10 feet, and dropped 1 foot. What is the current depth of the fish?
54. Which expression is represented by the model?
A -7 ft x{ÎÓ £ ä £ Ó Î { x
B -5 ft C -3 ft
F -5 + -1
H -5 + 1
D 7 ft
G -5 + 0
J
-5 + 4
55. CHEMISTRY The freezing point of oxygen is 219 degrees below zero on the Celsius scale. Use an integer to express this temperature. (Lesson 2-1) Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7)
56. age and family size
57. temperature and sales of mittens
ALGEBRA Find the solution of each equation from the list given. (Lesson 1-5) 58. 18 - n = 12; 3, 6, 30
59. 7a = 49; 7, 42, 343
PREREQUISITE SKILL Evaluate each expression if a = 6, b = 10, and c = 3. (Lesson 1-3) 60. a + 19 90 Chapter 2 Integers
61. 2b - 6
62. ab - ac
63. 3a - (b + c)
Learning Mathematics Vocabulary Some words used in mathematics are also used in English and have similar meanings. For example, in mathematics add means to combine. The meaning in English is to join or unite. Some words are used only in mathematics. For example, addend means a number to be added to another. Some words have more than one mathematical meaning. For example, an inverse operation undoes the effect of another operation, and an additive inverse is a number that when added to a given number gives zero. The list below shows some of the mathematics vocabulary used in Chapters 1 and 2.
Vocabulary
Meaning
Examples
algebraic expression
an expression that contains at least one variable and at least one mathematical operation
2 + x, _c , 3b
evaluate
to find the value of an expression
2+5=7
simplify
to find a simpler form of an expression
3b + 2b = 5b
integer
a whole number, its additive inverse, or zero
-3, 0, 2
factor
a number that is multiplied by another number
3(4) = 12 3 and 4 are factors.
product
the result of multiplying
3(4) = 12 ← product
quotient
the result of dividing two numbers
12 _ = 3 ← quotient
dividend
the number being divided
12← _ =3
dividend
divisor
the number being divided into another number
12 _ =3 ←
divisor
coordinate
a number that locates a point
(5, 2)
4
4 4
4
Reading to Learn 1. Name two of the words above that are also used in everyday English. Use the Internet, a dictionary, or another reference to find their everyday definition. How do the everyday definitions relate to the mathematical definitions? 2. Name two words above that are used only in mathematics. 3. Name two words above that have more than one mathematical meaning. List their meanings. Reading Math Learning Mathematics Vocabulary
91
EXPLORE
2-3
Algebra Lab
Subtracting Integers You can also use algebra tiles to model subtraction of integers. Remember one meaning of subtraction is to take away.
ACTIVITY
Use algebra tiles to find each difference.
a. 7 - 4
b. -8 - (-3)
Place 7 positive tiles on the mat. Remove 4 positive tiles.
Place 8 negative tiles on the mat. Remove 3 negative tiles.
£
£
£
£
£
£
£
£
£
£ £
£
£
£
£
So, 7 - 4 = 3 So, -8 - (-3) = -5. d. -6 - 2
c. 5 - (-2) Place 5 positive tiles on the mat and then remove 2 negative tiles. However, there are 0 negative tiles. First you must add 2 zero pairs to the set.
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
Then remove the 2 negative tiles. £
Place 6 negative tiles on the mat. Remove 2 positive tiles. Since there are no positive tiles, add 2 zero pairs to the mat.
£
£
£
£
Then remove the 2 positive tiles. £
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£ £
So, 5 - (-2) = 7.
£
£
So, -6 - 2 = -8.
ANALYZE THE RESULTS Apply what you learned to find each difference. 1. 9 - 7
2. 5 - (-3)
3. 6 - (-3)
4. 1 - (-5)
5. 3 - (-9)
6. -8 - 3
7. -8 - (-1)
8. -1 - 4
9. MAKE A CONJECTURE Write a rule that will help you determine the sign of the difference of two integers. 92 Chapter 2 Integers
2-3
Subtracting Integers BrainPOP® pre-alg.com
Main Ideas
You can use a number line to subtract integers. The model below shows how to find 6 - 8.
• Subtract integers. • Evaluate expressions containing variables.
Step 1 Step 2
⫺8
Start at 0. Move 6 units right to show positive 6.
6
From there, move 8 units left to subtract positive 8.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
1 2
3 4 5 6
7
a. What is 6 - 8? b. What direction do you move to indicate subtracting a positive integer? c. What addition sentence is also modeled by the number line above?
Subtract Integers When you subtract 8, as shown on the number line above, the result is the same as adding -8. When you subtract 5, the result is the same as adding -5. These examples suggest a method for subtracting integers. additive inverses
6 - 8 = -2
additive inverses
6 + (-8) = -2
-3 - 5 = -8
-3 + (-5) = -8
Subtracting Integers Words
To subtract an integer, add its additive inverse.
Symbols
a - b = a + (-b)
EXAMPLE Subtracting a Positive Integer To subtract a positive integer, think about moving left on a number line from the starting integer. In Example 1a, start at 8, then move left 13. You’ll end at -5. In Example 1b, start at -4, then move left 10. You’ll end at -14.
Subtract a Positive Integer
Find each difference. a. 8 - 13 8 - 13 = 8 + (-13) To subtract 13, add -13. = -5
Simplify.
b. -4 - 10 -4 - 10 = -4 + (-10) To subtract 10, add -10. = -14
1A. 9 - 16
Extra Examples at pre-alg.com
Simplify.
1B. -5 - 11 Lesson 2-3 Subtracting Integers
93
Review Vocabulary inductive reasoning making a conjecture based on a pattern of examples or past events (Lesson 1-1)
In Example 1, you subtracted a positive integer by adding its additive inverse. Use inductive reasoning to see if the method also applies to subtracting a negative integer. Subtracting an Integer ↔
Adding Its Additive Inverse
2-2=0 2-1=1 2-0=2 2 - (-1) = ?
2 + (-2) = 0 2 + (-1) = 1 2+0=2 2+1=3
Continuing the pattern in the first column, 2 - (-1) = 3. The result is the same as when you add the additive inverse.
EXAMPLE
Subtract a Negative Integer
Find each difference. a. 7 - (- 3)
b. -2 - (-4)
7 - (- 3) = 7 + 3 = 10
To subtract -3, add 3.
2A. 12 - (-4)
Real-World Link The hottest place in the world is Dallol, Ethiopia. High temperatures average 94.3°F throughout the year.
add 4.
=2
2B. -6 - (-15)
WEATHER The table shows the record high and low temperatures in selected states as of a recent year. What is the range, or difference between the highest and lowest temperatures, for Virginia? Explore
-2 - (- 4) = -2 + 4 To subtract -4,
State
Lowest Highest Temperature (˚F) Temperature (˚F)
Utah
-69
117
Vermont
-50
105
Virginia
-30
110
Washington -48 You know the West Virginia -37 highest and lowest temperatures. You Source: The World Almanac need to find the range for Virginia’s temperatures.
118
Plan
To find the range, or difference, subtract the lowest temperature from the highest temperature.
Solve
110 - (-30) = 110 + 30 To subtract -30, add 30. = 140
Source: Scholastic Book of World Records
Add 110 and 30.
The range for Virginia is 140°.
Check
Think of a thermometer. The difference between 110° above zero and 30° below zero must be 110 + 30 or 140°. The answer appears to be reasonable.
3. WEATHER What is the range of temperatures for Washington? Personal Tutor at pre-alg.com
94 Chapter 2 Integers Victor Englebert
112
Evaluate Expressions You can use the rule for subtracting integers to evaluate expressions.
EXAMPLE
Evaluate Algebraic Expressions
a. Evaluate x - (-6) if x = 12. x - (-6) = 12 - (- 6) Write the expression. Replace x with 12. = 12 + 6
To subtract -6, add its additive inverse, 6.
= 18
Add 12 and 6.
b. Evaluate a - b + c if a = 15, b = 5, and c = -8. a - b + c = 15 - 5 + (-8) Replace a with 15, b with 5, and c with -8. = 10 + (-8)
Order of operations
=2
Add 10 and -8.
Evaluate each expression if = 7, m = -3, and n = -10. 4B. - m + n 4A. n -
Examples 1, 2 (pp. 93–94)
Example 3 (p. 94)
Example 4 (p. 95)
HOMEWORK
HELP
For See Exercises Examples 14–21 1 22–31 2 32, 33 3 34–45 4
Find each difference. 1. 8 - 11
2. 10 - 15
3. - 10 - 14
4. -9 - 3
5. 7 - (-10)
6. 16 - (-12)
7. -6 - (-4)
8. -2 - (-8)
9. -15 - (- 18)
10. ANIMALS A gopher begins at 7 inches below the surface of a garden and digs down another 9 inches. Find an integer that represents the gopher’s position in relation to the surface of the garden. ALGEBRA Evaluate each expression if x = 10, y = -4, and z = -15. 11. x - (-10)
12. y - x
13. x + y - z
14. 3 - 8
15. 4 - 5
16. 2 - 9
17. 9 - 12
18. -3 - 1
19. -5 - 4
20. -6 - 7
21. -4 - 8
22. 6 - (-8)
23. 4 - (-6)
24. 7 - (-4)
25. 9 - (-3)
26. -9 - (-7)
27. -7 - (-10)
28. -11 - (-12)
29. -16 - (-7)
30. 10 - 24
31. 48 - (-50)
Find each difference.
32. MONEY Suppose you deposited $25 into your checking account and wrote a check for $38. What was the change in your account balance? 33. GEOGRAPHY The highest point in California is Mt. Whitney, with an elevation of 14,494 feet. The lowest point is Death Valley, with an elevation of -282 feet. Find the difference in the elevations. Lesson 2-3 Subtracting Integers
95
ALGEBRA Evaluate each expression if x = -3, y = 8, and z = -12.
Real-World Link Consumers spent a total of $38.4 billion on their lawns and gardens in 2003. Source: The National Gardening Association
34. y - 10
35. 12 - z
36. 3 - x
37. z - 24
38. x - y
39. z - x
40. y - z
41. z - y
42. x + y - z
43. z - y + x
44. x - y - z
45. z - y - x
ANALYZE TABLES For Exercises 46 and 47, use the table. 46. Describe the change in the sales related to each gardening activity from 2002 to 2003. 47. What was the total change in sales related to these gardening activities from 2002 to 2003?
3ALES 2ELATED TO 'ARDENING !CTIVITIES IN MILLIONS
!CTIVITY
)NDOOR HOUSEPLANTS ,ANDSCAPING ,AWN CARE 4REE CARE 6EGETABLE GARDENING 7ATER GARDENING
BUSINESS The formula P = I - E relates profit P to income I and expenses E. One month a small business has income of $19,592 and expenses of $20,345. 48. What is the profit for the month? 49. What does a negative profit mean?
-ÕÀVi\ / i >Ì> >À`i} ÃÃV>Ì
Find each difference. 50. 125 - (-114)
51. -320 - (-106)
52. -2200 - (-3500)
ANALYZE TABLES The daily closing prices for a company’s stock during one week are shown in the table. Date
Nov. 3
Nov. 4
Nov. 5
Nov. 6
Nov. 7
Closing Price
$33.30
$30.59
$31.04
$31.97
$30.15
—
?
?
?
?
Change
EXTRA
PRACTICE
53. Find the change in the closing price since the previous day. 54. What is the difference between the highest and lowest changes in closing price?
See pages 764, 795. Self-Check Quiz at pre-alg.com
H.O.T. Problems
55. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would subtract integers. CHALLENGE Determine whether each statement is true or false. If false, give a counterexample. 56. Subtraction of integers is commutative. 57. Subtraction of integers is associative. 58. FIND THE ERROR José and Amy are finding 8 - (-2). Who is correct? Explain your reasoning. José 8 - (-2) = 8 + 2 = 10
Amy 8 - (- 2) = 8 + (-2) =6
59. OPEN ENDED Write examples of a positive and a negative integer and their additive inverses. 96 Chapter 2 Integers Colin Paterson/Getty Images
60. SELECT A TECHNIQUE Reiko is filling out her check register. Which technique(s) might Reiko use to find out if she spent more money than she had in her checking account? Justify your selection(s). Then use the technique(s) to find how much she has left in her account.
#HECK .O $ATE $ESCRIPTION
mental math 61.
0AYMENT $EPOSIT "ALANCE
0AYCHECK 3CHOOL BOOKS )NITIATION FEE 'RAPHING CALCULATOR
number sense
estimation
Writing in Math Use the information about subtracting integers on page 93 to explain how the addition and subtraction of integers are related.
62. The melting point of metal mercury is -39°C. The freezing point of alcohol is -114°C. How much warmer is the melting point of mercury than the freezing point of alcohol? A -153°C
C 75°C
B -75°C
D 153°C
63. The terms in a pattern are given in the table. What is the value of the fifth term? Term Value
F -7
1 13
2 8
3 3
G -5
4 -2
5 ?
H 5
J 7
64. OCEANOGRAPHY A submarine at 1300 meters below sea level descends an additional 1150 meters. What integer represents the submarine’s position with respect to sea level? (Lesson 2-2) 65. ALGEBRA Evaluate ⎪b⎥ - ⎪a⎥ if a = 2 and b = -4. (Lesson 2-1) ALGEBRA Translate each phrase into an algebraic expression. (Lesson 1-3) 66. a number divided by 5 68. the quotient of eighty-six and b
67. the sum of t and 9 69. s decreased by 8
Find the value of each expression. (Lesson 1-2)
70. 2 × (5 + 8) - 6
71. 96 ÷ (6 × 8) ÷ 2
PREREQUISITE SKILL Find each product. 73. 5 · 15 74. 8 · 12
75. 3 · 5 · 8
72. 17 - (21 + 13) ÷ 17
76. 4 · 9 · 12
Lesson 2-3 Subtracting Integers
97
CH
APTER
2
Mid-Chapter Quiz Lessons 2-1 through 2-3
1. MULTIPLE CHOICE Choose the integer between 2 and -1. (Lesson 2-1) A -3 B -0.5 C 1 D 2.5 Replace each with , or = to make a true sentence. (Lesson 2-1) 2. 9 4. -8
3. -3
-5
5. 2
-6
0 -4
15. SPACE During night, the average temperature on Mars is -140°F. During the day, the average temperature rises 208°F. What is the average daytime temperature on Mars? (Lesson 2-2) 16. ACCOUNTING A small company had the following profits and losses for a six-month period. How much did the company earn during this time period? (Lesson 2-2)
6. MULTIPLE CHOICE Refer to the number line. Which statement is true? (Lesson 2-1) $ "
#
!
x{ Î Ó £ ä £ Ó Î { x
Jan.
-$3674
Feb.
$4013
Mar.
-$1729
Apr.
-$1415
F ⎪B⎥ < ⎪C⎥
H B>C
May
$1808
G C>A
J ⎪D⎥ > ⎪A⎥
Jun.
-$547
7. TEMPERATURE Order the temperatures from least to greatest. (Lesson 2-1) Temperature (˚F)
Liquid helium
-452
Outer space
-457
Dry ice
-108
Source: The Sizesaurus
9. -5 + 11
10. -6 + 9 + (-8)
11. 12 + (-6) + (-15)
12. ⎪-33 + 19⎥
13. ⎪-23 + -20⎥
14. MULTIPLE CHOICE Which day had the greatest change in stock price? (Lesson 2-2) Day Mon.
Open Price
Close Price
$43.29
$48.55
Tues.
$48.55
$46.65
Wed. Thurs.
$46.65 $41.30
$41.30 $45.99
A Mon. B Tues. C Wed. D Thurs. 98 Chapter 2 Integers
18. -15 - 8
19. 25 - (-7)
20. -16 - (-11)
ALGEBRA Evaluate each expression if x = 5, y = -2, and z = -3. (Lesson 2-3) 21. x - y
Find each sum. (Lesson 2-2) 8. -5 + (-15)
17. 16 - 23
22. z - 6
23. x - y - z
24. WEATHER If the temperature is -9°F and it drops 5°F overnight, what is the new temperature? (Lesson 2-3) 25. ASTRONOMY The graph shows the highest and lowest points of three planets. (Lesson 2-3) i} ÌÉ i«Ì ®
Item
Find each difference. (Lesson 2-3)
ÕÌ>à >` 6>iÞà Óx]äää Óä]äää £x]äää £ä]äää xäää ä xäää £ä]äää £x]äää
Ó£]ÎÎÈ £ä]ÈÈn
nnxä
ÓnÈ ÇÓx
££]äÎÎ
>ÀÌ
6iÕÃ *>iÌÃ
>ÀÃ
-ÕÀVi\ v«i>Ãi°V
What is the range of each of the planets? Which planet has the greatest range?
EXPLORE
2-4
Algebra Lab
Multiplying Integers You can also use algebra tiles to model multiplication of integers. Remember that 2 × 3 means two sets of three items. So, you can show 2 × 3 by placing 2 sets of 3 positive tiles on a mat. Similarly, you can model 2 × (-3) by placing 2 sets of 3 negative tiles on the mat, as shown at the right.
£
£
£
£
£
£
Ó Î® È
If the first factor is negative, you will need to remove tiles from the mat.
ACTIVITY Step 1 The expression -2 × (-3) means to remove 2 sets of 3 negative tiles. To do this, first place 2 × 3 or 6 zero pairs on the mat. £
£
£
£
£
£
£
£
£
£
£
£
Step 2 Then remove 2 sets of 3 negative tiles from the mat. There are 6 positive tiles remaining. So, -2 × (-3) = 6. Animation pre-alg.com £
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£ È
Ó Î®
EXERCISES Use algebra tiles to model and find each product. 1. 6 × (-2)
2. 3 × (-5)
3. 3 × (-4)
4. 1 × (-8)
5. -4 × (-2)
6. -5 × (-2)
7. -7 × (-1)
8. -2 × (-2)
9. Explain the meaning of -2 × 3. Then find the product using algebra tiles. Use algebra tiles to model and find each product. 10. -4 × 2
11. -3 × 5
12. -2 × 6
13. -1 × 3
ANALYZE THE RESULTS 14. How are the operations -3 × 4 and 4 × (-3) the same? How do they differ? 15. MAKE A CONJECTURE Find a rule you can use to find the sign of the product of two integers given the sign of both factors. Explore 2- 4 Algebra Lab: Multiplying Integers
99
2-4
Multiplying Integers
Main Ideas • Multiply integers. • Simplify algebraic expressions.
The temperature drops 7°C for each 1 kilometer increase in altitude. A drop of 7°C is represented by -7. So, the temperature change equals the altitude times -7. a. Suppose the altitude is 4 kilometers. Write an expression to find the temperature change.
Altitude (km)
Altitude Rate of Change
Temperature Change (°C)
1
1(7)
7
2
2(7)
14
3
3(7)
21
…
…
…
11
11(7)
77
b. Use the pattern in the table to find 4(-7).
Multiply Integers Multiplication is repeated addition. So, 3(-7) means Reading Math
that -7 is used as an addend 3 times.
Parentheses Recall that a product can be written using parentheses. Read 3(-7) as 3 times negative 7.
3(-7) = (-7) + (-7) + (-7) = -21
-7 -21
-7
-7 -14
-7
7
By the Commutative Property of Multiplication, 3(-7) = -7(3). This example suggests the following rule. Multiplying Integers with Different Signs Words
The product of two integers with different signs is negative.
Examples 4(-3) = -12
EXAMPLE
-3(4) = -12
Multiply Integers with Different Signs
Find each product. a. 5(-6) 5(-6) = -30
The factors have different signs. The product is negative.
b. -4(16) -4(16) = -64 The factors have different signs. The product is negative.
1A. 7(-8) 100 Chapter 2 Integers
1B. -6(12) Extra Examples at pre-alg.com
The product of two positive integers is positive. What is the sign of the product of two negative integers? Use a pattern to find a rule. One positive and one negative factor: Negative product
Two negative factors: Positive product
(-4)(2)
= -8
(-4)(1)
= -4
(-4)(0)
=
(-4)(-1) =
4
(-4)(-2) =
8
+4 Each product is 4 more than the previous product.
+4 +4 +4
Multiplying Integers with the Same Sign Words
The product of two integers with the same sign is positive.
Examples 4(3) = 12
EXAMPLE
-4(-3) = 12
Multiply Integers with the Same Sign
Find each product. b. -4(-5)(-8)
a. -6(-12) -6(-12) = 72 The product is
-4(-5)(-8) = [(-4)(-5)](-8)
positive.
2A. -5(-11)
= 20(-8) = -160
2B. -3(-4)(-5)
A glacier receded at a rate of 300 feet per day. What was the glacier’s movement in 5 days? A -1500 ft Context Clues Read the problem. Try to picture the situation. Look for words that suggest mathematical concepts.
B -300 ft
C -60 ft
D 305 ft
Read the Test Item The word receded means moved backward, so the rate per day is represented by -300. Multiply 5 and -300 to find the movement in 5 days. Solve the Test Item 5(-300) = -1500
The product is negative.
The answer is A.
3. A scuba diver descended at a rate of 5 feet per minute. What was the scuba diver’s depth at 5 minutes? F -25 ft
G -10 ft
H 10 ft
J 25 ft
Personal Tutor at pre-alg.com Lesson 2-4 Multiplying Integers
101
Algebraic Expressions You can use the rules for multiplying integers to simplify and evaluate algeraic expressions.
EXAMPLE
Simplify and Evaluate Algebraic Expressions
a. Simplify -2x(3y). -2x(3y) = (-2)(x)(3)(y)
-2x = (-2)(x), 3y = (3)(y)
= (-2 · 3)(x · y) Commutative Property of Multiplication = -6xy
-2 · 3 = -6, x · y = xy
b. Evaluate 4ab if a = 3 and b = -5. 4ab = 4(3)(-5)
Replace a with 3 and b with -5.
= [4(3)](-5)
Associative Property of Multiplication
= 12(-5)
The product of 4 and 3 is positive.
= -60
The product of 12 and -5 is negative.
4A. Simplify -3(6y). 4B. Simplify -7a(3b). 4C. Evaluate 2rs if r = 5 and s = -10.
Examples 1, 2 (pp. 100–101)
Find each product. 1. -3 · 8
2. 5(-8)
3. -2(11)
4. 4 · 30
5. -7(-4)
6. -6 · -6
7. -4(-2)(-6)
8. 8(-3)(-5)
9. -5(-9)(-12)
10. FITNESS The table shows burned Calories per minute for a 120-pound person during different activities. What is the change in the number of Calories in a 120-pound person’s body if he runs for 20 minutes?
!CTIVITY
#ALORIES PER -INUTE
"ALLET $ANCING "ICYCLING MPH 'OLF CARRYING CLUBS (ANDBALL 2UNNING 3KATEBOARDING
-ÕÀVi\ / i ÌiÃÃ *>ÀÌiÀ iVÌ
Example 3 (p. 101)
11. MULTIPLE CHOICE The research submarine Alvin, used to locate the wreck of the Titanic, descended at a rate of about 100 feet per minute. Which integer describes the distance Alvin traveled in 5 minutes? A -500 ft
Example 4 (p. 102)
B -100 ft
C -20 ft
D 100 ft
ALGEBRA Simplify each expression. 12. -4 · 3x
13. 7(-3y)
14. -8c(-3d)
ALGEBRA Evaluate each expression. 15. -6h, if h = -20 102 Chapter 2 Integers
16. -4st, if s = -9 and t = 3
HOMEWORK
HELP
For See Exercises Examples 17–22 1 23–32 2 33–50 4 62–63 3
Find each product. 17. -3 · 4
18. -7 · 6
19. 4(-8)
20. 9 · (-8)
21. -12 · 3
22. 14(-5)
23. 6 · 19
24. 4(32)
25. -8(-11)
26. -15(-3)
27. -5(-4)(6)
28. 5(-13)(-2)
29. -7(-8)(-3)
30. -6(-8)(11)
31. 2(-8)(-9)(10)
32. 4(-7)(-4)(-12)
ALGEBRA Simplify each expression. 33. -5 · 7x
34. -8 · 12y
35. 6(-8a)
36. 5(-11b)
37. -7s(-8t)
38. -12m(-9n)
39. 2ab(3)(-7)
40. 3x(5y)(-9)
41. -4(-p)(-q)
42. -8(-11b)(-c)
43. 9(-2c)(3d)
44. -6j(3)(5k)
ALGEBRA Evaluate each expression. 45. -7n, if n = -4
46. 9s, if s = -11
47. ab, if a = 9 and b = 8
48. -2xy, if x = -8 and y = 5
49. -16cd, if c = 4 and d = -5
50. 18gh, if g = -3 and h = 4
51. ELEVATORS An elevator takes students from the ground floor of a building down to an underground parking garage. Where will the elevator be in relation to the ground floor after 5 seconds if it travels at a rate of 3 feet per second? 52. TRAVEL A driver depresses the brake pedal of her car and begins decelerating at a rate of 2.3 meters per second per second. How much will the car’s speed change if the brake is applied for 6 seconds? 53. ANALYZE GRAPHS Write the product that is modeled on the number line.
EXTRA
PRACTICE
See pages 764, 795. Self-Check Quiz at pre-alg.com
H.O.T. Problems
-5 -15
-5
-5
-12 -10 -8 -6 -4 -2
2
4
TIDES For Exercises 54 and 55, use the information below. In Wrightsville, North Carolina, during low tide, the beachfront in some places is about 350 feet from the ocean to the homes. High tide can change the width of a beach at a rate of -17 feet an hour. It takes 6 hours for the ocean to move from low to high tide. 54. What is the change in the width of the beachfront from low to high tide? 55. What is the distance from the ocean to the homes at high tide? 56. ALGEBRA Find the values that complete the table below for y = -4x. x
-2
-1
1
y
?
?
?
?
57. OPEN ENDED Give an example of three integers whose product is negative. 58. REASONING Calculate (-10)(5)(18)[7 + (-7)] mentally. Justify your answer. 59. CHALLENGE Positive integers A and C satisfy A(A - C) = 23. What is the value of C? Lesson 2-4 Multiplying Integers
103
The cost of a trip to a popular amusement park can be determined with integers. Visit pre-alg.com to continue work on your project.
60. SELECT A TOOL During a drought, the amount of water in a pond changes by -9 gallons per day due to evaporation. Which of the following tools might you use to find the number of days it takes for the amount of water in a pond to change by -108 gallons of water? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model
61.
real objects
calculator
Writing in Math Explain how the signs of factors and products are related. Include an explanation of why the product of a positive and a negative integer must be negative.
62. An airplane descends at a rate of 200 feet per minute. Write a multiplication equation that tells the altitude of the airplane after 2 minutes.
63. GRIDDABLE At 8:00 P.M., a temperature of 78°F was recorded. The temperature then changed at an average rate of -2°F per hour for a 15-hour period. What was the temperature in degrees Fahrenheit at 7:00 A.M.?
A -200(2) = -400 C 200(2) = 400 B 200(-2) = -400 D -200(-2) = 400
ALGEBRA Evaluate each expression if a = -2, b = -6, and c = 14. (Lesson 2-3) 64. a - c 65. a - b 66. c - a + b 67. b - a + c 68. SWIMMING Lincoln High School’s swim team finished the 4 × 100-meter freestyle relay in 5 minutes 18 seconds. Prospect High School’s swim team finished the race in 5 minutes 7 seconds. Find an integer that represents Lincoln’s finish compared to Prospect’s finish. (Lesson 2-3) Find each sum. (Lesson 2-2) 69. -10 + 8 + 4
70. -4 + (-3) + (-7)
71. 9 + (-14) + 2 y
Refer to the coordinate system. Write the ordered pair that names each point. (Lesson 1-6) 72. E
73. C
74. B
75. F
76. D
77. A
B
D C
A
E F
O
PREREQUISITE SKILL Find each quotient. 78. 40 ÷ 8 79. 90 ÷ 15 104 Chapter 2 Integers
80. 45 ÷ 3
x
81. 91 ÷ 7
EXPLORE
2-5
Algebra Lab
Dividing Integers You can model division by separating algebra tiles into equal-sized groups.
ACTIVITY Use positive or negative tiles to find each quotient. a. 10 ÷ 2
b. -15 ÷ 5
Place 10 positive tiles on the mat to represent 10.
Place 15 negative tiles on the mat to represent -15.
£
£
£
£
£
£
£
£
£
£
Separate the tiles into 2 equal-sized groups.
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
£
Separate the tiles into 5 equal-sized groups. £
£
£
£
£
£
£
£
£
£
£
£
£
£
£
There are 5 positive tiles in each of the 2 groups.
There are 3 negative tiles in each of the 5 groups.
So, 10 ÷ 2 = 5.
So, -15 ÷ 5 = -3.
EXERCISES Apply what you learned to find each quotient. 1. 12 ÷ 6
2. 16 ÷ 2
3. 14 ÷ 7
4. -8 ÷ 2
5. -9 ÷ 3
6. -6 ÷ 2
7. -16 ÷ 4
8. -5 ÷ 5
9. -10 ÷ 2
ANALYZE THE RESULTS For Exercises 10–12, study the quotients in Exercises 1–9. 10. When the dividend and the divisor are both positive, is the quotient positive or negative? How does this compare to the sign of a product when both factors are positive? 11. When the dividend is negative and the divisor is positive, is the quotient positive or negative? How does this compare to the sign of a product when one factor is positive and one is negative? 12. MAKE A CONJECTURE Write a rule that will help you determine the sign of the quotient of two integers. Explore 2-5 Algebra Lab: Dividing Integers
105
2-5
Dividing Integers
Main Ideas • Divide integers. • Find the average of a set of data.
You can find the product 3 × (-4) on a number line. To find the product, start at 0 and then move -4 units three times. -12
New Vocabulary
-4
-4
-4
mean -12
-10
-8
-6
-4
-2
a. What is the product 3 × (-4)? b. What division sentence is also shown on the number line? c. Draw a number line and find the product 5 × (-2). Then find the related division sentence.
Divide Integers You can find the quotient of two integers by using the related multiplication sentence. Think of this factor . . . to find this quotient.
-4 × 3 = -12 → -2 × 5 = -10 →
Reading Math Parts of a Division Sentence In a division sentence, like 15 ÷ 5 = 3, the number you are dividing, 15, is called the dividend. The number you are dividing by, 5, is called the divisor. The result, 3, is called the quotient.
-12 ÷ (-4) = 3 -10 ÷ (-2) = 5
In the division sentences -12 ÷ (-4) = 3 and -10 ÷ (-2) = 5, notice that the dividends and divisors are both negative. In both cases, the quotient is positive. negative dividend and divisor
-12 ÷ (-4) = 3
-10 ÷ (-2) = 5 positive quotient
You already know that the quotient of two positive integers is positive. 12 ÷ 4 = 3
10 ÷ 2 = 5
These and similar examples suggest the following rule for dividing integers with the same sign. Dividing Integers with the Same Sign Words
The quotient of two integers with the same sign is positive.
Examples -12 ÷ (-3) = 4
106 Chapter 2 Integers
12 ÷ 3 = 4
EXAMPLE
Divide Integers with the Same Sign
Find each quotient. 75 b. _
a. -32 ÷ (-8) -32 ÷ (-8) = 4
5 75 = 75 ÷ 5 5
The quotient is positive.
The quotient is positive.
= 15
-39 1B. _
1A. 35 ÷ 5
-3
What is the sign of the quotient of a positive and a negative integer? Look for a pattern in the following related sentences. Think of this factor … to find this quotient.
-4 × (-6) = 24 → 2 × (-9) = -18 →
24 ÷ (-4) = -6 -18 ÷ 2 = -9
Notice that the signs of the dividend and divisor are different. In both cases, the quotient is negative. different signs
24 ÷ (-4) = -6 -18 ÷ 2 = -9
negative quotient
different signs
These and other similar examples suggest the following rule.
Dividing Integers with Different Signs Words
The quotient of two integers with different signs is negative.
Examples
-12 ÷ 4 = -3
EXAMPLE
12 ÷ (-4) = -3
Divide Integers with Different Signs
Find each quotient. Check Your Work Always check your work after finding an answer. If -42 ÷ 3 = -14, does -14 × 3 = -42?
a. -42 ÷ 3 -42 ÷ 3 = -14 The quotient is negative.
2A. 63 ÷ (- 7) Extra Examples at pre-alg.com
48 b. _
-6 48 _ = 48 ÷ (-6) The quotient is negative. -6 = -8 Simplify.
-110 2B. _ 11
Lesson 2-5 Dividing Integers
107
EXAMPLE
Evaluate Algebraic Expressions
Evaluate ab ÷ (-4) if a = -6 and b = -8. ab ÷ (-4) = -6(-8) ÷ (-4)
Replace a with -6 and b with -8.
= 48 ÷ (-4) or -12 Simplify.
3. Evaluate 12y ÷ x if x = -6 and y = -3.
Mean (Average) Division is used in statistics to find the average, or mean, of a set of data. To find the mean of a set of numbers, find the sum of the numbers and then divide by the number of items in the data set.
WEATHER The windchill temperatures in degrees Fahrenheit for the first six days in January were -2, 8, 5, -9, -12, and -2. Find the mean temperature for the six days. Checking Reasonableness
-2 + 8 + 5 + (-9) + (-12) + (-2) -12 ___ =_ 6
The average must be between the greatest and least numbers in the set. Is the average in Example 4 reasonable?
6
= -2
Find the sum of the set of integers. Divide by the number in the set. Simplify.
The mean temperature is -2°F.
4. GOLF Linda has scores of -3, -2, 1, and 0 during 4 rounds of golf. Find the mean of her golf scores. Personal Tutor at pre-alg.com
Operations with Integers Examples
Words Adding Two Integers To add integers with the same sign, add their absolute values. Give the result the same sign as the integers. To add integers with different signs, subtract their absolute values. Give the result the same sign as the integer with the greater absolute value.
-5 + (-4) = -9
5+4=9
-5 + 4 = -1
5 + (-4) = 1
5 - 9 = 5 + (-9) or -4 5 - (-9) = 5 + 9 or 14
Subtracting Two Integers To subtract an integer, add its additive inverse. Multiplying Two Integers The product of two integers with the same sign is positive. The product of two integers with different signs is negative. Dividing Two Integers The quotient of two integers with the same sign is positive. The quotient of two integers with different signs is negative.
108 Chapter 2 Integers
5 · 4 = 20
-5 · (-4) = 20
-5 · 4 = -20
5 · (-4) = -20
20 ÷ 5 = 4
-20 ÷ (-5) = 4
-20 ÷ 5 = -4
20 ÷ (-5) = -4
Examples 1, 2 (p. 107)
Example 3 (p. 108)
Example 4 (p. 108)
HOMEWORK
HELP
For See Exercises Examples 10–15 1 16–21 2 22–27 3 28, 29 4
Find each quotient. 1. 88 ÷ 8
2. -20 ÷ (-5)
-36 3. _
4. -18 ÷ 6
70 5. _ -7
-81 6. _ 9
-4
ALGEBRA Evaluate each expression. 8. _s , if s = -45 and t = 5
7. x ÷ 4, if x = -52
t
9. WEATHER The low temperatures for 7 days in January in degrees Fahrenheit were -2, 0, 5, -1, -4, 2, and 0. Find the average for the 7-day period.
Find each quotient. 10. 54 ÷ 9
11. 45 ÷ 5
12. -27 ÷ (-9)
13. -64 ÷ (-8)
14. -72 ÷ (-9)
15. -60 ÷ (-6)
16. -77 ÷ 7
17. -300 ÷ 6
18. 480 ÷ (-12)
-150 19. _ 10
600 20. _ -20
-350 21. _ 70
ALGEBRA Evaluate each expression. x 22. _ , if x = -85
-5 24. _c , if c = -63 and d = -7 d
26. xy ÷ (-3), if x = 9 and y = -7
108 23. _ m , if m = -9
25. _s , if s = 52 and t = -4 t
27. ab ÷ 6, if a = -12 and b = -8
28. STATISTICS Find the mean of 4, -8, 9, -3, -7, 10, and 2. 29. BASKETBALL In their first five games, the Jefferson Middle School basketball team scored 46, 52, 49, 53, and 45 points. What was their average number of points per game? ENERGY For Exercises 30 and 31, use the information below. h+ The formula d = 65 - _ can be used to find degree days, where h is the 2 high and is the low temperature. 30. If Las Vegas, Nevada, had a high of 94° and a low of 80°, find the degree days. 31. If Charleston, South Carolina, had a high of 56° and a low of 32°, find the degree days. 32. RESEARCH Use the Internet or another resource to find the high and low temperature for your city for a day in January. Find the degree days. EXTRA
PRACTICE
See pages 764, 795. Self-Check Quiz at pre-alg.com
33. SPACE The surface temperature on Mercury at night can fall to -300°F. 5(F - 32)
Use the expression _, where F represents the temperature in degrees 9 Fahrenheit, to find the temperature on Mercury in degrees Celsius. Round to the nearest tenth. Lesson 2-5 Dividing Integers
109
H.O.T. Problems
34. OPEN ENDED Write an equation with three integers that illustrates dividing integers with different signs. 35. CHALLENGE Find values for x, y, and z, so that all of the following statements are true. • y > x, z < y, and x < 0
• z ÷ 2 and z ÷ 3 are integers.
• x ÷ z = -z
• x÷y=z
36. CHALLENGE Addition and multiplication are said to be closed for whole numbers, but subtraction and division are not. That is, when you add or multiply any two whole numbers, the result is a whole number. Which operations are closed for integers? 37.
Writing in Math Use the information about dividing integers on pages 106–107 to explain how dividing integers is related to multiplying integers. Illustrate your answer with two related multiplication and division sentences.
38. The table shows the sales of a computer chip manufacturer in two recent years. What is the average change in sales per year? Year
Sales (millions)
2005 2000
$115 $128
39. Pedro has quiz scores of 8, 7, 8, and 9. What is the lowest score he can get on the remaining quiz to have a final average (mean) score of at least 8? F
7
G 8 H 9
A -$13 million
C $2.6 million
B -$2.6 million
D $13 million
Find each difference or product. (Lessons 2-3 and 2-4) 40. -8 - (-25) 41. 75 - 114
J 10
42. 2ab · (-2)
43. (-10c)(5d)
44. ANIMALS The height of an adult giraffe is 3 times the height of a newborn giraffe. Given n, the height of a newborn giraffe, write an equation that can be used to find a, the height of an adult giraffe. (Lesson 1-5) 45. PATTERNS Find the next two numbers in the pattern 5, 4, 2, -1, …. (Lesson 1-1)
PREREQUISITE SKILL Use the coordinate plane to name the point for each ordered pair. (Lesson 1-6) 46. (1, 5) 47. (6, 2) 48. (4, 5)
y
C B
D E
A F
49. (0, 3)
G H O
110 Chapter 2 Integers
x
2-6
The Coordinate System
Main Ideas • Graph points on a coordinate plane. • Graph algebraic relationships.
New Vocabulary
A GPS, or Global Positioning System, can be used to find a location anywhere on Earth by identifying its latitude and longitude. Several cities are shown on the map below. For example, Brisbane, Australia, is located at approximately 30°S, 150°E. Èäc
Èäc
'REENWICH
quadrants
"EIJING
$ALLAS
Îäc
Îäc
äc
äc
Îäc-
"RISBANE
Îäc-
£xäc
£Óäc
äc
Èäc
Îäc
#APE 4OWN äc
Îäc7
Èäc7
äc7
£Óäc7
£xäc7
3ANTIAGO
a. Latitude is measured north and south of the equator. What is the latitude of Dallas? b. Longitude is measured east and west of the prime meridian. What is the longitude of Dallas? c. What does the location 32°N, 100°W mean?
Graph Points Latitude and longitude are a kind of coordinate system. The coordinate system, or coordinate plane you used in Lesson 1-6 can be extended to include points below and to the left of the origin. Review Vocabulary
4 3 2 1 4 321 O 1 2 P (4, 2) 3 4
1 2 3 4x
The x-axis extends to the right and left of the origin. Notice that the numbers to the left of zero on the x-axis are negative.
Recall that a point graphed on the coordinate system has an x-coordinate and a y-coordinate. The dot at the ordered pair (-4, -2) is the graph of point P. x-coordinate
y-coordinate
(-4, -2)
冦
Coordinate System a coordinate plane formed by the intersection of two number lines that meet at right angles at their zero points (Lesson 1-6)
origin
The y-axis extends above and below the origin. Notice that the numbers below zero on the y-axis are negative.
y
ordered pair Lesson 2-6 The Coordinate System
111
EXAMPLE
Write Ordered Pairs
Write the ordered pair that names each point. a. A Ordered Pairs Notice that the axes in an ordered pair (x, y) are listed in alphabetical order.
A
The x-coordinate is -3. The y-coordinate is 2. The ordered pair is (-3, 2).
y
-4 -3-2-1 O 1 2 3 4 x -1 B -2 C D -3 -4 5
b. B The x-coordinate is 4. The y-coordinate is -2. The ordered pair is (4, -2).
1A. C
4 3 2 1
1B. D
The x-axis and the y-axis separate the coordinate plane into four regions, called quadrants. The axes and points on the axes are not located in any of the quadrants. The quadrants are named I, II, III, and IV.
The coordinates are (negative, positive).
The coordinates are (negative, negative).
EXAMPLE Interactive Lab pre-alg.com
II
4 y 3 2 1
-4 -3-21 O -1 III -2 -3 -4
I
The coordinates are (positive, positive).
1 2 3 4x
IV
The coordinates are (positive, negative).
Graph Points and Name the Quadrant
Graph and label each point on a coordinate plane. Name the quadrant in which each point lies. y a. E(2, 4) ( Start at the origin. Move 2 units right. Then move 4 units up and draw a dot. Point E(2, 4) is in Quadrant I. b. F(-3, -2) Start at the origin. Move 3 units left. Then move 2 units down and draw a dot. Point F(-3,-2) is in Quadrant III.
4 3 2 1
-4 -3-2-1 O -1 -2 ( ) F -3, -2 -3 -4
E 2, 4)
G (4, 0) 1 2 3 4x
c. G(4, 0) Start at the origin. Move 4 units right. Since the y-coordinate is 0, the point lies on the x-axis. Point G(4, 0) is not in any quadrant.
2A. H(4, -3) 112 Chapter 2 Integers
2B. J(0, -2)
2C. I(-1, 4)
Reading Math Coordinate System Coordinate plane, coordinate grid, and coordinate graph are other names for the coordinate system.
Graph Algebraic Relationships You can use a coordinate graph to show relationships between two numbers.
EXAMPLE
Graph an Algebraic Relationship
The sum of two numbers is 5. If x represents the first number and y represents the second number, make a table of possible values for x and y. Graph the ordered pairs and describe the graph. First, make a table. Choose values for x and y that have a sum of 5. Then graph the ordered pairs on a coordinate plane. x+y=5 y
(x, y)
2
3
(2, 3)
1
4
(1, 4)
5
(0, 5)
-1
6
(-1, 6)
-2
7
(-2, 7)
x
y
O
x
The points on the graph are in a line that slants downward to the right. The line crosses the y-axis at y = 5.
3. The difference of two numbers is 4. If x represents the first number and y represents the second number, make a table of possible values for x and y. Graph the ordered pairs and describe the graph. Personal Tutor at pre-alg.com
Example 1 (p.112)
Name the ordered pair for each point graphed at the right. 1. A
2. C
3. G
4. K
y
A
D B
C
O
Example 2 (p.112)
Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 5. J(3, -4)
6. K(-2, 2)
7. L(0, 4)
8. M(-1, -2)
x
G F
H
K
9. GEOMETRY Graph points A(-4, 3), B(1, 3), C(1, 2), and D(-4, 2) on a coordinate plane and connect them to form a rectangle. Name the quadrant in which each point is located. Example 3 (p.113)
10. ALGEBRA Make a table of values and graph six ordered integer pairs where x + y = 3. Describe the graph.
Extra Examples at pre-alg.com
Lesson 2-6 The Coordinate System
113
HOMEWORK
HELP
For See Exercises Examples 11–20 1 21–32 2 33–38 3
Name the ordered pair for each point graphed at the right. 11. R
12. G
13. M
14. B
15. V
16. H
17. U
18. W
19. A
y
R H G
V W x
O
T
U
M B
A
20. T
Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 21. A(4, 5)
22. H(0, -3)
23. M(4, -2)
24. B(-5, -5)
25. S(2, -5)
26. F(-4, 0)
27. E(0, 3)
28. K(-5, 1)
29. G(5, 0)
30. C(6, -1)
31. D(0, 0)
32. R(-3, 5)
ALGEBRA Make a table of values and graph six sets of ordered integer pairs for each equation. Describe the graph. 33. x + y = 4
34. x - y = -2
35. y = 2x
36. y = -2x
37. y = x + 2
38. y = x - 1
Graph each point. Then connect the points in alphabetical order and identify the figure. 39. A(0, 6), B(4, -6), C(-6, 2), D(6, 2), E(-4, -6), F(0, 6)
Reading Math Vertex, Vertices A vertex of a triangle is a point where two sides of a triangle meet. Vertices is the plural of vertex.
40. A(5, 8), B(1, 13), C(5, 18), D(9, 13), E(5, 8), F(5, 6), G(3, 7), H(3, 5), I(7, 7), J(7, 5), K(5, 6), L(5, 3), M(3, 4), N(3, 2), P(7, 4), Q(7, 2), R(5, 3), S(5, 1) GEOMETRY On a coordinate plane, draw triangle ABC with vertices at A(3, 1), B(4, 2), and C(2, 4). Then graph and describe each new triangle formed in Exercises 41–44. 41. Multiply each coordinate of the vertices in triangle ABC by 2. 42. Multiply each coordinate of the vertices in triangle ABC by -1. 43. Add 2 to each coordinate of the vertices in triangle ABC. 44. Subtract 4 from each coordinate of the vertices in triangle ABC. 45. RESEARCH Find a map of your school and draw a coordinate grid on the map with the library as the center. Locate the cafeteria, principal’s office, your math classroom, gym, counselor’s office, and the main entrance on your grid. Write the coordinates of these places. How can you use these points to help visitors find their way around your school? Graph and label each point on a coordinate plane. 46. A(-6.5, 3)
EXTRA
PRACTICE
47. B(-2, -5.75)
48. C(4.1, -1)
49. D(-3.4, 1.5)
See page 765, 795.
50. ALGEBRA Graph eight ordered integer pairs where ⎪x⎥ > 3. Describe the graph.
Self-Check Quiz at pre-alg.com
51. ALGEBRA Graph all ordered integer pairs that satisfy the condition ⎪x⎥ < 4 and ⎪y⎥ < 3.
114 Chapter 2 Integers
H.O.T. Problems
52. OPEN ENDED Name two ordered pairs whose graphs are not located in one of the four quadrants. 53. FIND THE ERROR Keisha says that if you interchange the coordinates of any point in Quadrant I, the new point would be in Quadrant I. Jason says the new point would be in Quadrant III. Who is correct? Explain your reasoning. CHALLENGE If the graph of A(x, y) satisfies the given condition, name the quadrant in which point A is located. 54. x > 0, y > 0
55. x < 0, y < 0
56. x < 0, y > 0
57. NUMBER SENSE Graph eight sets of integer coordinates that satisfy ⎪x⎥ + ⎪y⎥ > 3. Describe the location of the points. 58.
Writing in Math Use the information on page 111 to explain how a coordinate plane is used to locate places on Earth. Include an explanation of how coordinates can describe a location and how latitude and longitude are related to the x- and y-axes on a coordinate plane.
For Exercises 59 and 60, refer to the graph at the right. 59. On the coordinate plane, what are the coordinates of the point that shows the location of the library? A (4, -2)
C (4, 2)
B (-2, -4)
D (-4, -2)
y Pool Park O x Library
Grocery Store
60. On the coordinate plane, what location has coordinates (5, -2)? F park
H library
G school
J
School
store
Find each quotient. (Lesson 2-5)
61. -24 ÷ (-8)
62. 105 ÷ (-5)
63. -400 ÷ (-50)
ALGEBRA Evaluate each expression if f = -9, g = -6, and h = 8. (Lesson 2-4) 64. -5fg
65. 2gh
66. -10fh
67. WEATHER In the newspaper, Ruben read that the low temperature for the day was expected to be -5ºF and the high temperature was expected to be 8ºF. What was the difference in the expected high and low temperatures? (Lesson 2-3) ALGEBRA Simplify each expression. (Lesson 1-4) 68. (a + 8) + 6
69. 4(6h)
70. (n · 7) · 8
71. (b · 9) · 5
72. (16 + 3y) + y
73. 0(4z) Lesson 2-6 The Coordinate System
115
CH
APTER
2
Study Guide and Review
wnload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
Key Concepts Integers and Absolute Value
(Lesson 2-1)
• Numbers on a number line increase as you move from left to right. • The absolute value of a number is the distance the number is from zero on the number line.
absolute value (p. 80) additive inverse (p. 88) coordinate (p. 79) inequality (p. 79) integers (p. 78) mean (p. 108) negative number (p. 78) opposites (p. 88) quadrants (p. 112)
Adding and Subtracting Integers (Lessons 2-2 and 2-3)
• To add integers with the same sign, add their absolute values. Give the result the same sign as the integers.
Vocabulary Check
• To add integers with different signs subtract their absolute values. Give the result the same sign as the integer with the greater absolute value.
Complete each sentence with the correct term. Choose from the list above. 1. A(n)_______ is a number less than zero.
• To subtract an integer, add its additive inverse.
Multiplying and Dividing Integers (Lessons 2-4 and 2-5)
• The product or quotient of two integers with the same sign is positive. • The product or quotient of two integers with different signs is negative.
The Coordinate Plane
(Lesson 2-6)
• The x-axis and the y-axis separate the coordinate plane into four quadrants. • The axes and points on the axes are not located in any of the quadrants.
2. The number that corresponds to a point on the number line is called the ________. 3. An integer and its opposite are also called _________ of each other. 4. The four regions separated by the axes on a coordinate plane are called ________. 5. The set of _________ includes positive whole numbers, their opposites, and zero. 6. The _________ of a number is the distance the number is from zero on the number line. 7. A(n) _______ is a mathematical sentence containing < or >. 8. To find the _______ of a set of numbers, find the sum of the numbers and then divide by the number of items in the data set. 9. Two numbers with the same absolute values but different signs are _______.
116 Chapter 2 Integers
Vocabulary Review at pre-alg.com
Mixed Problem Solving
For mixed problem-solving practice, see page 795.
Lesson-by-Lesson Review 2–1
Integers and Absolute Value
(pp. 78–83)
Replace each with , or = to make a true sentence. 11. -3 -3 10. 8 -8 12. -2
13. -12
-21
Evaluate each expression. 15. ⎪25⎥ 14. ⎪-32⎥ 17. ⎪-8⎥ + ⎪-14⎥
16. -⎪15⎥
Example 1 Replace with , or = in -3 2 to make a true sentence. ⫺4⫺3 ⫺2⫺1 0 1 2 3 4
Since -3 is to the left of 2, -3 < 2. Example 2 Evaluate ⎪-4⎥. 4 units
18. BASEBALL CARDS Jamal traded away 7 shortstop cards for 5 outfielder cards. Find an integer that represents the change in the number of cards Jamal had after the trade.
2–2
Adding Integers
The graph of -4 is 4 units from 0. So, ⎪-4⎥ = 4.
(pp. 86–90)
Example 3 Find -3 + (-4).
Find each sum. 19. -6 + (-3)
20. -4 + (-1)
21. -2 + 7
22. 4 + (-8)
23. 4 + 7 + (-3)
24. -9 + 6 + (-8)
25. GOLF A golfer’s scores for the last five weeks are -5, +7, -2, -4, and +5. What is the sum of his scores?
2–3
⫺5⫺4⫺3 ⫺2 ⫺1 0 1 2
Subtracting Integers
-3 + (-4) = -7
The sum is negative.
Example 4 Find 5 + (-2). 5 + (-2) = 3 The sum is positive.
(pp. 93–97)
Find each difference. 26. 4 - 9 27. -3 - 5 28. 7 - (-2)
29. -1 - (-6)
30. -7 - 8
31. 6 - 10
32. ELEVATORS The postal carrier entered the elevator on floor 15. She rode down 6 floors. Then she rode up 10 floors and got off. What floor was she on when she left the elevator?
Example 5 Find -5 - 2. -5 - 2 = -5 + (-2) To subtract 2, add -2. = -7 Example 6 Find 8 - (-4). 8 - (-4) = 8 + 4 To subtract -4, add 4. = 12
Chapter 2 Study Guide and Review
117
CH
A PT ER
2 2–4
Study Guide and Review
Multiplying Integers
(pp. 100–104)
Example 7 Find 6(-4).
Find each product. 33. -9(5)
34. 11(-6)
35. -4(-7)
36. -3(-16)
37. SNOWBOARDING For each trick he completes incorrectly in the half pipe event, Kurt receives -3 points. If Kurt incorrectly completes five tricks, what is his score?
2–5
Dividing Integers
40. -36 ÷ 9
41. 88 ÷ (-4)
42. RACING The number of seconds Elena is behind the leader for the first five legs of the bicycle race is shown. What is her average time behind the leader? +32 s, +5 s, +10 s, +8 s, +12 s
The Coordinate System
45. K(-1, 3)
46. R(3, 0)
47. GAMES The coordinate plane represents a board game. Name the quadrant in which each player’s game piece is located.
-8(-2) = 16 The factors have the same sign, so the product is positive.
Example 9 Find -30 ÷ (-5). -30 ÷ (-5) = 6
The dividend and divisor have the same sign, so the quotient is positive.
Example 10 Find 27 ÷ (-3). 27 ÷ (-3) = -9 The dividend and divisor have different signs, so the quotient is negative.
Example 11 Graph and label F(5, -3) on a coordinate plane. Name the quadrant in which the point is located. Point F(5, -3) is in Quadrant IV.
Y
"
X
&
Y
"Üi
`Þ
X
"
iÀÞ >L
118 Chapter 2 Integers
Example 8 Find -8(-2).
(pp. 111–115)
Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. 43. A(4, 3) 44. J(-2, -5)
iÞVi
The factors have different signs, so the product is negative.
(pp. 106–110)
Find each quotient. 38. -14 ÷ (-2) 39. -52 ÷ (-4)
2–6
6(-4) = -24
CH
A PT ER
2
Practice Test
Write two inequalities using the numbers in each sentence. Use the symbols < and >. 1. -5 is less than 2. 2. 12 is greater than -15. 3. MULTIPLE CHOICE A scuba diver records her depth in the lake every minute. Choose the group of depths that is listed in order from least to greatest.
23. WEATHER The table shows the low temperatures during one week in Anchorage, Alaska. Find the average low temperature for the week. Day Temperature (°F)
S
M
T
W
T
F
S
-12
3
-7
-4
1
-2
A -13 ft, -12 ft, -9 ft, -3 ft, -1 ft, -5 ft B -5 ft, -3 ft, -1 ft, -9 ft, -12 ft, -13 ft C -12 ft, -13 ft, -3 ft, -1 ft, -9 ft, -5 ft D -13 ft, -12 ft, -9 ft, -5 ft, -3 ft, -1 ft 4. FOOTBALL During the first play of the game, the Brownville Tigers football team lost seven yards. On each of the next two plays, an additional four yards were lost. Express the total yards lost at the end of the first three plays as an integer. Find each sum or difference.
ALGEBRA Evaluate each expression if a = -5, b = 3, and c = -10. 24. ab - c
25. c ÷ a
bc 26. _ a -6
27. 4c + ⎪a⎥
28. MULTIPLE CHOICE A vertex of a triangle is a point where two sides of the triangle meet. Which ordered pair is not a vertex of ABC?
5. -4 + (-8)
6. -9 + 15
F (-1, 1)
7. 12 + (-15)
8. 14 + (-7) + -11
G (2, -3)
9. 4 - 13 11. -6 - (-10)
10. 8 - (-6)
H (1, 2)
12. -14 - (-7)
J (-1, -1)
13. STOCK MARKET On Thursday, a company’s stock closed at $67.24. On Friday, it closed at $64.27. What was the change in the closing price?
Y
"
#
Graph and label each point on a coordinate plane. Name the quadrant in which each point is located.
Find each product or quotient.
29. D(-2, 4)
30. E(3, -4)
14. 6(-8)
15. -9(8)
31. F(-1, -3)
32. G(3, 2)
16. -7(-5)
17. 2(-4)(11)
18. 54 ÷ (-9)
19. -64 ÷ (-4)
20. -250 ÷ 25
21. -144 ÷ (-6)
22. SWIMMING POOL The water in a swimming pool drains at a rate of 24 gallons per minute. Describe the change in the amount of water in the swimming pool after 1 hour.
Chapter Test at pre-alg.com
X
"
!
33. MULTIPLE CHOICE Suppose Elan’s home represents the origin on a coordinate plane. If Elan leaves his home and walks two miles west and then four miles north, what is his current location as an ordered pair? A (-2, 4)
C (-2, -4)
B (2, 4)
D (4, -2)
Chapter 2 Practice Test
119
CH
A PT ER
2
Standardized Test Practice Cumulative, Chapters 1–2
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
5. What are the coordinates of the center of the circle below? £ £ " £ Ó Î { x È Ç
1. A scuba diver descends at a rate of 40 feet per minute. Which equation shows how far the scuba diver moves in 2 minutes? C 40(2) = 80 A -40(-2) = 80 B -40(2) = -80 D 40(-2) = -80 2. On Wednesday, the low temperature in Fairbanks, Alaska was -6°F, and the high temperature was 14°F. How much warmer was the high temperature than the low temperature? F -20°F H 8°F G -8°F J 20°F 3. GRIDDABLE Michael had $45 in his savings account at the beginning of the week. He made a withdrawal of $22 to buy a video game on Tuesday, and he made a deposit of $25 on Friday when he received some money for his birthday. How much money in dollars did Michael have in the account at the end of the week if he made no other withdrawals or deposits? 4. Tyrone’s long distance phone bills were $21.35, $11.14, $22.82, and $33.05 over the past four months. He estimated that the phone bill would cost $80 over these four months. Which statement best describes how reasonable his estimate is? A Less than the actual amount because he rounded to the nearest $10 B Less than the actual amount because he rounded to the nearest $100 C More than the actual amount because he rounded to the nearest $10 D More than the actual amount because he rounded to the nearest $100
120 Chapter 2 Integers
F G H J
Y £ Ó Î { x È ÇX
(-3, 2) (-2, 3) (3, -2) (2, -3)
6. GRIDDABLE What is the twelfth term of the pattern given by the expression below where n is the term number? 3(n - 5)
7. Last week, Traci wrote checks for $32, $58, and $14. She also made two deposits totaling $189. What other information is needed in order to find the current balance in Traci’s account? A The amount of each deposit. B Traci’s balance last week. C To whom Traci wrote checks last week. D Traci’s deposit when she opened the account.
8. Of the six books in a mystery series, four have 200 pages and two have 300 pages. Which expression represents the total number of pages in the series? F 200 + 300 H 4(200) + 2(300) G 6(200 + 300) J 8(200 + 300)
Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
9. A pattern of equations is shown below. 1% of 2,000 = 20 2% of 1,000 = 20 4% of 500 = 20 8% of 250 = 20 Which statement best describes this pattern? A When the percent is doubled and the other number is doubled, the answer is 20. B When the percent is doubled and the other number is halved, the answer is 20. C When the percent is increased by 2 and the other number remains the same, the answer is 20. D When the percent remains the same and the other number is increased by 2, the answer is 20. 10. Tonya wants to order a roast beef sandwich, a medium order of fries, and a medium drink. How much money will she save by ordering a Daily Special #2?
11. Before the last game of the season, Amy had scored a total of 58 goals. She scored 4 goals in the final game, making her season average 3.1 goals per game. To find the total number of games that Amy played, first find the sum of 58 and 4, and then— A add the sum to 3.1. B subtract 3.1 from 58. C multiply the sum by 3.1. D divide the sum by 3.1.
Question 12 When answering open-ended items on standardized tests, follow these steps: 1. Read the item carefully. 2. Show all of your work. You may receive points for items that are only partially correct. 3. Check your work.
Pre-AP Record your answers on a sheet of paper. Show your work. 12. On graph paper, graph the points A(4, 2), B(-3, 7), and C(-3, 2). Connect the points to form a triangle. a. Add 6 to the x-coordinate of each coordinate pair. Graph and connect the new points to form a new figure. Is the new figure the same size and shape as the original triangle? Describe how the size, shape, and position of the new triangle relate to the size, shape, and position of the original triangle. b. If you add -6 to each original x-coordinate, and graph and connect the new points to create a new figure, how will the position of the new figure relate to that of the original one?
F $1.22 G $0.84 H $0.78 J $0.38
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
Go to Lesson...
2- 4
2-3
2-2
1-1
1-6
2-3
2-2
1-1
1-1
1-1
1-1
2-6
Chapter 2 Standardized Test Practice
121
3
Equations
•
Select and use appropriate operations to solve problems and justify solutions.
•
Use graphs, tables, and algebraic representations to make predictions and solve problems.
Key Vocabulary area (p. 163) formula (p. 162) like terms (p. 129) sequence (p. 158) simplest form (p. 130)
Real-World Link Skyscrapers Rising 630 feet to the top, the Gateway Arch in St. Louis, Missouri, is 130 feet higher than Mount Rushmore in Black Hills, South Dakota, 75 feet higher than the Washington Monument, and 25 higher than the Seattle Space Needle.
quations Make this Foldable to help you organize information about expressions and equations. Begin ith five sheets of 812’’ × 11’’ paper.
1 Stack 5 sheets of paper 3 4 inch apart.
2 Roll up the bottom edges. All tabs should be the same size.
3 Crease and staple
4 Label the tabs with
along the fold.
topics from the chapter.
122 Chapter 3 Equations Gibson Stock Photography
>«ÌiÀ Î
µÕ>ÌÃ
Ûi *À«iÀÌÞ
£° ÃÌÀLÕÌ Ã } Ý«ÀiÃà Ӱ -« vÞ Î° µÕ>ÌÃ\
X Ã\ {° µÕ>Ì
µÕ>ÌÃ x° /Ü-Ìi« Õ>ÌÃ È° 7ÀÌ} µ >ÌÃ
µÕ Ç° -iµÕiVià n° ÀÕ>Ã Ü ° ,iÛi
GET READY for Chapter 3 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Find each product. (Lesson 2-4) 1. 2(-3) 2. -4(3) 3. -5(-2)
4. -4 · 6
5. -11 · -8
6. 9 · (-4)
7. STOCK MARKET The price of a stock decreased $2.05 each day for five consecutive days. What was the total change in value of the stock over the five-day period? (Lesson 2-4)
Write each subtraction expression as an addition expression. (Lessons 2-3) 8. 5 - 7 9. 6 - 10 10. -13 - 9
11. 11 - 10
12. 15 - 6
13. -19 - 10
Example 1 Find 7(-2).
7(-2) = -14
The factors have different signs, so the product is negative.
Example 2 Find -5 · -9.
-5 · -9 = 45 The factors have the same sign, so the product is positive.
Example 3 Write 8 - 12 as an addition expression.
8 - 12 = 8 + (-12) To subtract 12, add -12. = -4
Simplify.
14. MONEY Student Council spent $178 on decorations and $110 on snacks for the dance. Write an addition expression for the amount remaining in the dance budget if Student Council initially had $593. (Lessons 2-3)
Find each sum. (Lesson 2-2) 15. 6 + (-9) 16. -8 + 4 17. 4 + (-4)
18. 7 + (-10)
19. -13 + (-8)
20. -11 + 12
Example 4 Find -5 + 7.
-5 + 7 = 2 Subtract | -5 | from | 7 |. The sum is positive because | 7 | > | -5 |.
21. CAVERNS A tour group began 26 feet underground. During their tour, they descended 15 feet more and then ascended 19 feet. Express their current depth as an integer. (Lesson 2-2)
Chapter 3 Get Ready for Chapter 3
123
3-1
The Distributive Property BrainPOP at pre-alg.com
Main Ideas • Use the Distributive Property to write equivalent numerical expressions. • Use the Distributive Property to write equivalent algebraic expressions.
To find the area of a rectangle, multiply the length and width. You can find the total area of the blue and yellow rectangles in two ways. Method 1
Method 2
Put them together. Add the lengths. Then multiply. 4
+
Separate them. Multiply to find each area. Then add. 4
2
2
New Vocabulary equivalent expressions
3
3
3(4 + 2) = 3 · 6 Add. = 18 Multiply.
+
3
3 · 4 + 3 · 2 = 12 + 6 Multiply. = 18 Add.
a. Draw a 2-by-5 and a 2-by-4 rectangle. Find the total area in two ways. b. Draw a 4-by-4 and a 4-by-1 rectangle. Find the total area in two ways. c. Draw any two rectangles that have the same width. Find the total area in two ways. d. What did you notice about the total area in each case?
Distributive Property The expressions 3(4 + 2) and 3 · 4 + 3 · 2 are equivalent expressions because they have the same value, 18. This example shows how the Distributive Property combines addition and multiplication. Vocabulary Link Distribute Everyday Use to deliver to each member of a group Distributive Math Use property that allows you to multiply a number by a sum
Distributive Property Words
To multiply a number by a sum, multiply each number inside the parentheses by the number outside the parentheses.
Symbols
a(b + c) = ab + ac
Examples 3(4 + 2) = 3 · 4 + 3 · 2
(b + c)a = ba + ca (5 + 3)2 = 5 · 2 + 3 · 2
You can use the Distributive Property to evaluate numerical or algebraic expressions. 124 Chapter 3 Equations
EXAMPLE
Use the Distributive Property
Use the Distributive Property to write each expression as an equivalent expression. Then evaluate the expression. a. 2(6 + 4)
b. (8 + 3)5
2(6 + 4) = 2 · 6 + 2 · 4 = 12 + 8 Multiply. = 20 Add.
(8 + 3)5 = 8 · 5 + 3 · 5 = 40 + 15 Multiply. = 55 Add.
1A. (6 + 3)4
1B. 4(2 + 9)
AMUSEMENT PARKS A one-day pass to an amusement park costs $40. A round-trip bus ticket to the park costs $5. a. Write two equivalent expressions to find the total cost of a one-day pass and a bus ticket for 15 students. Method 1 Find the cost for 1 person, then multiply by 15. 15($40 + $5) 15 times the cost for 1 person Method 2 Find the cost of 15 passes and 15 tickets. Then add. 15($40) + 15($5) cost of 15 passes + cost of 15 tickets b. Find the total cost.
Real-World Link Attendance at U.S. amusement parks increased 22% in the 1990s. In 2004, about 328 million people attended these parks.
15($40 + $5) = 15($40) + 15($5) Distributive Property = $600 + $75 Multiply. = $675 Add. The total cost is $675. You can check your results by evaluating 15($45).
2. FOOD A spaghetti dinner costs $10 and a slice of pie costs $2. Write two equivalent expressions to find the total cost of a spaghetti dinner and a slice of pie for each member of a family of 4. Then find the total cost.
Source: International Association of Amusement Parks and Attractions
Personal Tutor at pre-alg.com
Algebraic Expressions You can also model the Distributive Property by using algebra tiles and variables. 4HE MODEL SHOWS X 4HERE ARE GROUPS OF X
3EPARATE THE TILES INTO GROUPS OF X AND GROUPS OF
X Î Ó
X X
£ £
X £ £
£ £
Ó
X X
Î
Ó
£ £
£ £
£ £
2(x + 3) = 2x + 2 · 3 = 2x + 6 The expressions 2(x + 3) and 2x + 6 are equivalent expressions because for every value of x, these expressions have the same value. Extra Examples at pre-alg.com AP/Wide World Photos
Lesson 3-1 The Distributive Property
125
EXAMPLE
Simplify Algebraic Expressions
Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 3(x + 1)
b. (y + 4)5
3(x + 1) = 3x + 3 · 1 = 3x + 3
(y + 4)5 = y · 5 + 4 · 5
3A. 2(a + 5)
EXAMPLE
= 5y + 20
Simplify.
Simplify.
3B. (b + 6)3
Simplify Expressions with Subtraction
Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 2(x - 1) 2(x - 1) = 2[x + (-1)]
Look Back To review subtraction expressions, see Lesson 2–3.
Rewrite x - 1 as x + (-1).
= 2x + 2(-1)
Distributive Property
= 2x + (-2)
Simplify.
= 2x - 2
Definition of subtraction
b. -3(n - 5) -3(n - 5) = -3[n + (-5)]
Rewrite n - 5 as n + (-5).
= -3n + (-3)(-5) Distributive Property = -3n + 15
4A. 4(d - 3)
Example 1 (p. 125)
(p. 125)
Examples 3, 4 (p. 126)
4B. -7(e - 4)
Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. 1. 5(7 + 8)
Example 2
Simplify.
2. 2(9 + 1)
3. (2 + 4)6
4. (3 + 6)4
MONEY For Exercises 5 and 6, use the following information. Suppose you work in a grocery store 4 hours on Friday and 5 hours on Saturday. You earn $6.25 an hour. 5. Write two different expressions to find your wages. 6. Find the total wages for that weekend. ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 7. 4(x + 3) 11. 8(y - 2)
126 Chapter 3 Equations
8. 8(m + 4) 12. 9(a - 10)
9. (n + 2)3 13. -6(x - 5)
10. (p + 4)5 14. -3(s - 7)
HOMEWORK
HELP
For See Exercises Examples 15–26 1 27, 28 2 29–36 3 37–44 4
Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. 15. 2(6 + 1) 19. (9 + 2)4 23. -3(9 - 2)
16. 5(7 + 3) 20. (8 + 8)2 24. -2(8 - 4)
17. (4 + 6)9 21. 7(3 - 2) 25. -5(8 - 4)
18. (4 + 3)3 22. 6(8 - 5) 26. -5(10 - 3)
2 7. MOVIES One movie ticket costs $7, and one small bag of popcorn costs $3. Write two equivalent expressions for the total cost of four movie tickets and four bags of popcorn. Then find the cost. 28. SPORTS A volleyball uniform costs $15 for the shirt, $10 for the pants, and $8 for the socks. Write two equivalent expressions for the total cost of 12 uniforms. Then find the cost. ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 29. 33. 3 7. 41.
2(x + 3) (x + 3)4 3(x - 2) (r - 5)6
30. 34. 38. 42.
3 1. 35. 39. 43.
5(y + 6) (y + 2)10 9(m - 2) (x - 3)12
ANALYZE GRAPHS For Exercises 45–47, use the double bar graph.
7(y + 8) (2 + x)5 15(s - 3) (a - 6)(-5)
Annual Fashion Spending $2200 $1964
$2000
Average Spending per Teen
45. Find the total amount spent on average by two male teenagers and two female teenagers on fashion products in 2003. 46. Find the total amount spent on average by three female teenagers in both 2002 and 2003 on fashion products. 47. Did teen spending on fashion products increase? How do you know? Explain.
32. 36. 40. 44.
3(n + 1) (3 + y)6 8(z - 3) -2(z - 4)
$1800 $1600 $1400
$1342
$1200 $890
$1000
$834
$800 $600 $400 $200 0
2002 Female
2003 Male
Source: www.piperjaffray.com
ALGEBRA Use the Distributive Property to write each expression as an equivalent algebraic expression. 48. 2(x + y) 5 1. 4(j - k) EXTRA
PRACTICE
See pages 765, 796. Self-Check Quiz at pre-alg.com
49. 3(a + b) 52. 10(r - s)
50. (e + f)(-5) 53. (u - w)(-8)
MENTAL MATH Find each product mentally. Example 15 · 12 = 15(10 + 2) Think: 12 is 10 + 2. = 150 + 30 or 180 Distributive Property 54. 7 · 14
55. 8 · 23
56. 9 · 32
57. 16 · 11
Lesson 3-1 The Distributive Property
127
58. THEME PARKS Admission to an amusement park is $41.99 for adults and $26.99 for children. The Diego family has a coupon for $10 off each ticket. Write an expression for the cost for x adults and y children.
H.O.T. Problems
59. OPEN ENDED Write an equation using three integers that is an example of the Distributive Property. 60. FIND THE ERROR Julia and Catelyn are using the Distributive Property to simplify 3(x + 2). Who is correct? Explain your reasoning. Catelyn 3(x + 2) = 3x + 6
Julia 3(x + 2) = 3x + 2
61. CHALLENGE Is 3 + (x · y) = (3 + x) · (3 + y) a true statement? If so, explain your reasoning. If not, give a counterexample. 62.
Writing in Math Explain how rectangles can be used to show the Distributive Property.
63. A ticket to a baseball game costs t dollars. A soft drink costs s dollars. Which expression represents the total cost of a ticket and soft drink for p people? A pst
C t(p + s)
B p + (ts)
D p(t + s)
64. Which equation is always true? F 5(a + b) = 5a + b G 5(ab) = (5a)(5b) H 5(a + b) = 5(b + a) J 5(a + 0) = 5a + 5
ALGEBRA The table shows several solutions of the equation x + y = 4. (Lesson 2-6) 65. Graph the ordered pairs on a coordinate plane. 66. Describe the graph. -4y
x+y=4 x
y
(x, y)
-1
5
(-1, 5)
1 2
3 2
(1, 3) (2, 2)
67. ALGEBRA Evaluate x if x = 2 and y = -3. (Lesson 2-5) 68. FITNESS Jake ran x miles on Monday, y miles on Tuesday, and z miles on Wednesday. Write an expression for the average number of miles Jake ran. (Lesson 1-2)
PREREQUISITE SKILL Write each subtraction expression as an addition expression. (Lesson 2-3) 69. 5 - 3 70. -8 - 4 71. 10 - 14 72. 8 - (-6) 128 Chapter 3 Equations
3-2
Simplifying Algebraic Expressions
Main Idea • Use the Distributive Property to simplify algebraic expressions.
New Vocabulary term coefficient like terms constant simplest form simplifying an expression
In a set of algebra tiles, X represents the variable x,
represents the
£
integer 1, and £ represents the integer -1. You can use algebra tiles to represent expressions. You can also sort algebra tiles by their shapes and group them. The tiles below represent the expression 2x + 3 + 3x + 1. On the right, the algebra tiles have been sorted and combined. X TILES £
X
X
£
£
X
X
X
X
£
X
X
X
X
£
£
£
£
TILES ÓX
Î
ÎX
xX
£
{
Therefore, 2x + 3 + 3x + 1 = 5x + 4. Model each expression with algebra tiles or a drawing. Then sort them by shape and write an expression represented by the tiles. a. 3x + 2 + 4x + 3
b. 2x + 5 + x
c. 4x + 5 + 3
d. x + 2x + 4x
Simplify Expressions When plus or minus signs separate an algebraic expression into parts, each part is a term. The numerical part of a term that contains a variable is called the coefficient of the variable. Four terms
2x + 8 + x + 8 2 is the coefficient of 2x.
Vocabulary Link Constant Everyday Use unchanging Math Use a fixed value in an expression
1 is the coefficient of x because x = 1x.
Like terms are terms that contain the same variables, such as 2n and 5n or 6xy and 4xy. A term without a variable is called a constant. Constant terms are also like terms. Like terms
5y + 3 + 2y + 8y Constant
Lesson 3-2 Simplifying Algebraic Expressions
129
Rewriting a subtraction expression using addition will help you identify the terms of an expression.
EXAMPLE
Identify Parts of Expressions
Identify the terms, like terms, coefficients, and constants in the expression 3x - 4x + y - 2. 3x - 4x + y - 2 = 3x + (– 4x) + y + (–2)
Definition of subtraction
= 3x + (– 4x) + 1y + (–2) Identity Property The terms are 3x, – 4x, y, and –2. The like terms are 3x and – 4x. The coefficients are 3, –4, and 1. The constant is –2.
1. Identify the terms, like terms, coefficients, and constants in the expression 9a – 2a + 3b – 5.
An algebraic expression is in simplest form if it has no like terms and no parentheses. When you use the Distributive Property to combine like terms, you are simplifying the expression.
EXAMPLE
Simplify Algebraic Expressions
Simplify each expression. a. 6n + 3 + 2n Equivalent Expressions To check whether 6n + 2n and 8n are equivalent expressions, substitute any value for n and see whether the expressions have the same value.
6n and 2n are like terms. 6n + 3 + 2n = 6n + 2n + 3 Commutative Property = (6 + 2)n + 3 Distributive Property = 8n + 3
Simplify.
b. 3x - 5 - 8x + 6 3x and -8x are like terms. -5 and 6 are also like terms. 3x - 5 - 8x + 6 = 3x + (-5) + (-8x) + 6
Definition of subtraction
= 3x + (-8x) + (-5) + 6
Commutative Property
= [3 + (-8)]x + (-5) + 6
Distributive Property
= -5x + 1
Simplify.
c. m + 3(n + 4m) m + 3(n + 4m) = m + 3n + 3(4m) Distributive Property
2A. 4x + 6 - 3x 130 Chapter 3 Equations
= m + 3n + 12m
Associative Property
= 1m + 3n + 12m
Identity Property
= 1m + 12m + 3n
Commutative Property
= (1 + 12)m + 3n
Distributive Property
= 13m + 3n
Simplify.
2B. 2m + 3 - 7m - 4
2C. 4(q + 8p) + p Extra Examples at pre-alg.com
BASEBALL CARDS Suppose your brother has 15 more baseball cards in his collection than you have. Write an expression in simplest form that represents the total number of cards in both collections. Words Variables Expression
Source: CMG Worldwide
plus
number of your brother’s cards
Let x = number of cards you have. Let x + 15 = number of cards your brother has. x + (x + 15)
x + (x + 15) = (x + x) + 15 = (1x + 1x) + 15 = (1 + 1)x + 15 = 2 x + 15
Real-World Link Honus Wagner is considered by many to be baseball’s greatest all-around player. In July, 2000, one of his baseball cards sold for $1.1 million.
number of your cards
Associative Property Identity Property Distributive Property Simplify.
The expression 2x + 15 represents the total number of cards, where x is the number of cards you have.
3. STAMPS Matt and Lola both collect stamps. Lola has 25 more stamps in her collection than Matt has. Write an expression in simplest form that represents the total number of stamps in both collections. Personal Tutor at pre-alg.com
Example 1 (p. 130)
Identify the terms, like terms, coefficients, and constants in each expression. 2. 2m - n + 6m
3. 4y - 2x - 7
4. 6a + 4 + 2a
5. x + 9x + 3
6. 9y + 8 - 8
7. 3x + 2y + 4y
8. 6c + 4 + c + 8
9. 2x - 5 - 4x + 8
10. x + 3(x + 4y)
11. 8e - 4(2f + 5e)
1. 4x + 3 + 5x + y Example 2 (p. 130)
12. 5 - 3(y + 7)
(p. 131)
13. MONEY You have saved some money. Your friend has saved $20 more than you. Write an expression in simplest form that represents the total amount of money you and your friend have saved.
HELP
Identify the terms, like terms, coefficients, and constants in each expression.
Example 3
HOMEWORK
Simplify each expression.
For See Exercises Examples 14–19 1 20–34 2 35–38 3
14. 3 + 7x + 3x + x
15. y + 3y + 8y + 2
16. 2a + 5c - a + 6a
17. 5c - 2d + 3d - d
18. 6m - 2n + 7
19. 7x - 3y + 3z - 2 Lesson 3-2 Simplifying Algebraic Expressions
Kit Kittle/CORBIS
131
Simplify each expression. 20. 2x + 5x
21. 7b + 2b
22. y + 10y
23. 5y + y
24. 2a + 3 + 5a
25. 4 + 2m + m
26. 2y + 8 + 5y + 1
27. 8x + 5 + 7 + 2x
28. 5x - 3x
29. 10b - 2b
30. 4y - 5y
31. r - 3r
32. 8 + x - 5x
33. 6x + 4 - 7x
34. 2x + 3 - 3x + 9
For Exercises 35–38, write an expression in simplest form that represents the total amount in each situation. 35. SCHOOL SUPPLIES You bought 5 folders that each cost x dollars, a calculator for $45, and a set of pens for $3. 36. SHOPPING Suppose you buy 3 shirts that each cost s dollars, a pair of shoes for $50, and jeans for $30. 37. FASHION Your friend Natasha has y pairs of shoes. Her sister has 5 fewer pairs. 38. BABY-SITTING Alicia earned d dollars baby-sitting. Her friend earned twice as much. You earned $2 less than Alicia’s friend earned. Simplify each expression. Real-World Link In a recent survey, 10% of students in grades 6–12 reported that most of their spending money came from baby-sitting. Source: USA WEEKEND
39. 6m + 2n + 10m
40. -2y + x + 3y
41. c + 2(d - 5c)
42. 3(b + 2) + 2b
43. 5(x + 3) + 8x
44. -3(a + 2) - a
45. -2(x + 3) + 2x
46. 4x - 4(2 + x)
47. 8a - 2(a - 7)
GEOMETRY You can find the perimeter of a geometric figure by adding the measures of its sides. Write an expression in simplest form for the perimeter of each figure. 2x ⫹ 1 48. 49. 3x
EXTRA
PRACTIICE
H.O.T. Problems
x
x 2x ⫹ 1
See pages 765, 796. Self-Check Quiz at pre-alg.com
5x
4x
Simplify to make the calculation as easy as possible. 50. 16 · (-31) + 16 · 32
51. 72(38) + (-72)(18)
52. OPEN ENDED Write an expression in simplest form containing three terms. One of the terms should be a constant. 53. FIND THE ERROR Koko and John are simplifying the expression 5x - 4 + x + 2. Who is correct? Explain your reasoning. Koko 5x - 4 + x + 2 = 6x - 2
John 5x - 4 + x + 2 = 5x - 2
54. Which One Doesn’t Belong? Identify the algebraic expression that does not belong with the other three. Explain your reasoning. -6(x - 2) 132 Chapter 3 Equations Mary Kate Denny/PhotoEdit
x + 12 - 7x
-x - 5x + 12
-6x - 12
CHALLENGE You use deductive reasoning when you base a conclusion on mathematical rules or properties. Indicate the property that justifies each step that was used to simplify 3(x + 4) + 5(x + 1). 55. 3(x + 4) + 5(x + 1) = 3x + 12 + 5x + 5 56.
= 3x + 5x + 12 + 5
57.
= 3x + 5x + 17
58.
= 8x + 17
59.
Writing in Math Explain how algebra tiles can be used to simplify an algebraic expression. Illustrate your reasoning with an example.
60. The perimeter of DEF is 4x + 3y. What is the measure of the third side of the triangle? A -2x + 2y
$ xX ÎY
B 2x + 2y C x-y
X {Y
&
%
D -x + 2y
61. You spend x minutes reading a book on Saturday. On Sunday, you spend 35 more minutes reading than you did on Saturday. Which expression represents the total amount of time spent reading the book on Saturday and Sunday? F 2x + 35
H 2x - 35
G x + 35
J x - 35
ALGEBRA Use the Distributive Property to write each expression as an equivalent expression. (Lesson 3-1) 62. 3(a + 5)
63. -2(y - 8)
64. 7(d - 10)
65. -3(x - 1)
66. Name the quadrant in which P(-5, -6) is located. (Lesson 2-6) 67. CRUISES The table shows the number of people who took a cruise in various years. Make a scatter plot of the data. (Lesson 1-7)
9EAR .UMBER MILLIONS -ÕÀVi\ #RUISE ,INES )NTERNATIONAL !SSOCIATION
3(4a - 3b) b-4
68. ALGEBRA What is the value of _ if a = 6 and b = 7? (Lesson 1-3) 69. DECORATING A wallpaper roll contains a sheet that is 40 feet long and 18 inches wide. What is the minimum number of rolls of wallpaper needed to cover 500 square feet of wall space? (Lesson 1-1)
PREREQUISITE SKILL Find each sum. (Lesson 2-2) 70. -5 + 4
71. -8 + (-3)
72. 10 + (-1)
Lesson 3-2 Simplifying Algebraic Expressions
133
EXPLORE
3-3 AND 3-4
Algebra Lab
Solving Equations Using Algebra Tiles ACTIVITY 1 In a set of algebra tiles, X represents the variable x,
£
represents the
integer 1, and £ represents the integer -1. You can use algebra tiles and an equation mat to model equations.
£
X
£
£
£
X Î
£
£
£
£
X
£
£
£
X Ó
x
£
When you solve an equation, you are trying to find the value of x that makes the equation true. The following example shows how to solve x + 3 = 5 using algebra tiles.
£
X
£
£
X Î
£
£
£
X
x
X
Remove the same number of 1-tiles from each side of the mat until the x-tile is by itself on one side.
£
X
Model the equation.
£
X
£
£
The number of tiles remaining on the right side of the mat represents the value of x.
Ó
Therefore, x = 2. Since 2 + 3 = 5, the solution is correct.
EXERCISES Use algebra tiles or a drawing to model and solve each equation. 1. 3 + x = 7 2. x + 4 = 5 3. 6 = x + 4 4. 5 = 1 + x 134 Chapter 3 Equations
ACTIVITY 2
Animation at pre-alg.com
Some equations are solved by using zero pairs. You may add or subtract a zero pair from either side of an equation mat without changing its value. The following example shows how to solve x + 2 = -1 by using zero pairs.
£
X
£
X Ó
£
£
£
£
£
X Ó Ó®
X
£
X
£
Model the equation. Notice it is not possible to remove the same kind of tile from each side of the mat.
£
£ £
X
£
£
Add 2 negative 1-tiles to the left side of the mat to make zero pairs. Add 2 negative 1-tiles to the right side of the mat.
£ Ó®
£
£
£
£
Remove all of the zero pairs from the left side. There are 3 negative 1-tiles on the right side of the mat.
Î
Therefore, x = -3. Since -3 + 2 = -1, the solution is correct.
EXERCISES Use algebra tiles or a drawing to model and solve each equation. 5. x + 2 = -2 6. x - 3 = 2 7. 0 = x + 3 8. -2 = x + 1
ACTIVITY 3 The equation 2x = -6 is modeled using more than one x-tile. Arrange the tiles into equal groups to match the number of x-tiles. X
X ÓX
£
£
£
£
£
£ È
X
£
£
£
£
£
£
X X
Î
Therefore, x = -3. Since 2(-3) = -6, the solution is correct.
EXERCISES Use algebra tiles or a drawing to model and solve each equation. 9. 3x = 3 10. 2x = -8 11. 6 = 3x 12. -4 = 2x Explore 3-3 and 3-4 Algebra Lab: Solving Equations Using Algebra Tiles
135
3-3 Main Ideas • Solve equations by using the Subtraction Property of Equality. • Solve equations by using the Addition Property of Equality.
New Vocabulary inverse operation equivalent equations
Solving Equations by Adding or Subtracting On the balance at the right, the paper bag contains a certain number of blocks. (Assume that the paper bag weighs nothing.) a. Without looking in the bag, how can you determine the number of blocks in the bag? b. Explain why your method works.
Solve Equations by Subtracting The equation x + 4 = 7 is a model of the situation above. You can use inverse operations to solve the equation. Inverse operations “undo” each other. For example, to undo the addition of 4 in the expression x + 4, you would subtract 4. To solve the equation x + 4 = 7, subtract 4 from each side. x+4=7 x+4-4=7-4 Subtract 4 from the left side of the equation to isolate the variable.
x+0=3 x=3
Subtract 4 from the right side of the equation to keep it balanced.
The solution is 3. You can use the Subtraction Property of Equality to solve any equation like x + 4 = 7.
Subtraction Property of Equality Words
If you subtract the same number from each side of an equation, the two sides remain equal.
Symbols
For any numbers a, b, and c, if a = b, then a - c = b - c.
Examples
5=5 5-3=5-3 2=2
x+2=3 x+2-2=3-2 x=1
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
136 Chapter 3 Equations
The equations x + 4 = 7 and x = 3 are equivalent equations because they have the same solution, 3. When you solve an equation, you should always check to be sure that the first and last equations are equivalent.
EXAMPLE
Solve Equations by Subtracting
Solve x + 8 = -5. Check your solution and graph it on a number line. x + 8 = -5 x + 8 - 8 = -5 - 8 x + 0 = -13 x = -13 Checking Equations It is always wise to check your solution. You can often use arithmetic facts to check the solutions of simple equations.
Write the equation. Subtract 8 from each side. 8 - 8 = 0, -5 - 8 = -13 Identity Property; x + 0 = x
To check your solution, replace x with -13 in the original equation. CHECK
x + 8 = -5 -13 + 8 -5 -5 = -5
Write the equation. Check to see whether this sentence is true. The sentence is true.
The solution is -13. To graph it, draw a dot at -13 on a number line. £{ £Î £Ó ££ £ä n
Ç
Solve each equation. Check your solution and graph it on a number line. 1A. 4 = x + 10 1B. 16 + z = 14
Solve Equations by Adding Some equations can be solved by adding the same number to each side. This uses the Addition Property of Equality. Addition Property of Equality Words
If you add the same number to each side of an equation, the two sides remain equal.
Symbols
For any numbers a, b, and c, if a = b, then a + c = b + c.
Examples
6=6 6+3=6+3 9=9
x-2=5 x-2+2=5+2 x=7
If an equation has a subtraction expression, first rewrite the expression as an addition expression. Then add the additive inverse to each side.
EXAMPLE
Solve Equations by Adding
Solve y - 7 = -25. y - 7 = -25 y + (-7) = -25 y + (-7) + 7 = -25 + 7 y + 0 = -25 + 7 y = -18 The solution is -18.
Write the equation. Rewrite y - 7 as y + (-7). Add 7 to each side. Additive Inverse Property; (-7) + 7 = 0. Identity Property; y + 0 = y Check your solution.
Solve each equation. 2A. -20 = y - 13 Extra Examples at pre-alg.com
2B. -115 + b = -84 Lesson 3-3 Solving Equations by Adding or Subtracting
137
Jessica downloaded 54 songs onto her digital music player. This is 17 less than the number of songs Kaela downloaded earlier. Which equation can be used to find the number of songs Kaela downloaded onto her digital music player? Key Words When translating words to equation, look for key words that indicate operations. The phrase “less than” can indicate subtraction or an inequality.
A x - 17 = 54 B x + 17 = 54
C 17 - x = 54 D -54 = 17 + x
Read the Test Item Translate the verbal sentence into an equation. Solve the Test Item Words
Jessica downloaded 17 less songs than
Variable
Let = the number of songs Kaela downloaded.
Equation
54 = x - 17
So, the equation 54 = x - 17 or x - 17 = 54 can be used to find the number of songs Kaela downloaded. This is choice A.
3. During the night, the temperature dropped 14° to -9°F. Which equation can be used to find the temperature at the beginning of the night? F -9 + x = -14
H -9 - x = 14
G 14 - x = -9
J x - 14 = -9
Personal Tutor at pre-alg.com
SLEDDING Use the information at the left. Write and solve an equation to find the distance of the Northern Route of the Iditarod Trail Sled Dog Race. The Southern Route is 49 miles longer than the Northern Route.
Real-World Link There are two different routes for the Iditarod Trail Sled Dog Race. During the odd years, the race takes place on the 1161-mile Southern Route. This is 49 miles longer than the Northern Route that takes place during the even years. Source: iditarod.com
138 Chapter 3 Equations AP/Wide World Photos
Let d = the distance of the Northern Route. 1161= d + 49 Write the equation. 1161 - 49 = d + 49 - 49 Subtract 49 from each side. 1112 = d Simplify. CHECK
1161 = d + 49 1161 1112 + 49 1161 = 1161
Write the equation. Check to see whether this statement is true. The statement is true.
The Northern Route is 1112 miles long.
4. BUILDINGS The Jefferson Memorial in Washington, D.C., is 129 feet tall. This is 30 feet taller than the Lincoln Memorial. Write and solve an equation to find the height of the Lincoln Memorial.
2. w + 4 = -10
3. 16 = y + 20
4. n - 8 = 5
5. k - 25 = 30
6. r - 4 = -18
7. MULTIPLE CHOICE A video store sells a DVD for $12 more than it pays for it. If the selling price of the DVD is $19, which equation can be used to find how much the store paid for the DVD? A x + 19 = -12
For See Exercises Examples 9–26 1, 2 27, 28 4 43, 44 3
ALGEBRA Solve each equation. Check your solution and graph it on a number line. 9. y + 7 = 21
10. x + 5 = 18
1 1. m + 10 = -2
12. x + 5 = -3
13. a + 10 = -4
14. t + 6 = -9
15. y + 8 = 3
16. 9 = 10 + b
17. k - 6 = 13
18. r - 5 = 10
19. 8 = r - 5
20. 19 = g - 5
21. x - 6 = -2
22. y - 49 = -13
23. -15 = x - 16
24. -8 = t - 4
25. 23 + y = 14
26. 59 = s + 90
27. ELECTIONS In the 2004 presidential election, Georgia had 15 electoral votes. That was 19 votes fewer than the number of electoral votes in Texas. Write and solve an equation to find the number of electoral votes in Texas. 28. WEATHER The difference between the record high and low temperatures in Charlotte, North Carolina, is 109˚F. The record low temperature was -5˚F. Write and solve an equation to find the record high temperature. 29. RESEARCH Use the Internet or another source to find record temperatures in your state. Use the data to write a problem. ANALYZE GRAPHS For Exercises 30 and 31, use the graph and the following information. Tokyo’s population is 10 million greater than New York City’s population. Los Angeles’ population is 2 million less than New York City’s population. 30. Write two different equations to find New York City’s population. 31. Solve the equations to find the population of New York City.
Most Populous Urban Areas 28
?
18
18
18 16 14
Shanghai
HELP
8. FUND-RAISING Jim sold 43 candles to raise money for a class trip. This is 15 less than the number Diana sold. Write and solve an equation to find the number of candles Diana sold.
Los Angeles
HOMEWORK
D 12 - x = 19
Sao Paulo
(p. 138)
C x + 12 = 19
Bombay
Example 4
B 19 + x = 12
Mexico City
(p. 138)
1. x + 14 = 25
New York City
Example 3
ALGEBRA Solve each equation. Check your solution and graph it on a number line.
Tokyo
(p. 137)
Population (millions)
Examples 1, 2
Cities Source: infoplease.com
Lesson 3-3 Solving Equations by Adding or Subtracting
139
ALGEBRA Solve each equation. Check your solution.
PRACTICE
32. a - 6.1 = 3.4
33. 14.8 + x = - 20.1
34. 17.6 = y + 11.5
See pages 766, 796.
35. p - (-13.35) = -19.72
36. -52.23 + b = 40.04
37. z - 37.98 = 65.21
Self-Check Quiz at pre-alg.com
38. ALGEBRA If a number x satisfies x + 4 = -2, find the numerical value of -3x - 2.
EXTRA
H.O.T. Problems
39. OPEN ENDED Write two equations that are equivalent. Then write two equations that are not equivalent. Justify your reasoning. 40. SELECT A TECHNIQUE Jaime’s golf score was -9 today. She decreased her score by 5 strokes from yesterday. Which of the following techniques might you use to determine what her golf score was yesterday? Justify your selection(s). Then use the technique(s) to solve the problem. computer
draw a model
real objects
41. CHALLENGE Write two equations in which the solution is -5. 42.
Writing in Math Formulate a problem situation for the equation x + 7 = 20.
The table shows the five nearest stars to Earth, excluding the Sun. Star
Distance (light-years)
Proxima Centauri
4.22
Alpha Centauri A
4.40
Alpha Centauri B
4.40
Barnard’s Star
5.94
Wolf 359
7.79
43. Which equation will best help you find how much closer Proxima Centauri is to Earth than Barnard’s Star? A x - 5.94 = 4.22 C 5.94 + x = 4.22 B x + 4.22 = 5.94 D 5.94 + 4.22 = x 44. GRIDDABLE How many light years closer is Alpha Centauri B to Earth than Wolf 359?
ALGEBRA Simplify each expression. (Lessons 3-1 and 3-2) 45. -2(x + 5)
46. (t + 4)3
47. -4(x - 2)
48. 6z - 3 - 10z + 7
49. 2(x + 6) + 4x
50. 3 - 4(m + 1)
51. GEOLOGY The width of a beach is changing at a rate of -9 inches per year. How long will it take for the width of the beach to change -4.5 feet? (Lesson 2-5) 52. MONEY Xavier opened a checking account with a deposit of $200. During the next week, he wrote checks for $65, $83, and $28 and made a deposit of $50. Write an addition expression for this situation and find the balance in his account. (Lesson 2-2)
PREREQUISITE SKILL Divide. (Lesson 2–5) 53. -100 ÷ 10 54. 50 ÷ (-2) 140 Chapter 3 Equations
55. -49 ÷ (-7)
72 56. _ -8
3-4
Solving Equations by Multiplying or Dividing
Main Ideas • Solve equations by using the Division Property of Equality. • Solve equations by using the Multiplication Property of Equality.
An exchange rate allows people to exchange one currency for another. In Mexico, about 11 pesos can be exchanged for $1 of U.S. currency, as shown in the table.
U.S. Value ($)
Number of Pesos
1
11(1) ⴝ 11
2
11(2) ⴝ 22
3
11(3) ⴝ 33
4
11(4) ⴝ 44
In general, if we let d represent the number of U.S. dollars and p represent the number of pesos, then 11d = p.
a. Suppose lunch in Mexico costs 77 pesos. Write an equation to find the cost in U.S. dollars. b. How can you find the cost in U.S. dollars?
Solve Equations by Dividing The equation 11x = 77 is a model of the relationship described above. To undo the multiplication operation in 11x, you would divide by 11. To solve the equation 11x = 77, divide each side by 11. 11x = 77 Divide the left side of the equation by 11 to undo the multiplication 11 · x.
11x 77 _ = _ 11 11
1x = 7 x = 7
Divide the right side of the equation by 11 to keep it balanced.
The solution is 7. You can use the Division Property of Equality to solve any equation like 11x = 77.
Division Property of Equality Words
When you divide each side of an equation by the same nonzero number, the two sides remain equal.
b _ For any numbers a, b, and c, where c ≠ 0, if a = b then _ c = c. Examples 14 = 14 3x = -12 a
Symbols
14 _ 14 _ =
3x _ -12 _ =
2=2
x = -4
7
7
3
3
Lesson 3-4 Solving Equations by Multiplying or Dividing
141
EXAMPLE
Solve Equations by Dividing
Solve 5x = -30. Check your solution and graph it on a number line. 5x = -30
Write the equation.
5x -30 = 5 5
Divide each side by 5 to undo the multiplication in 5 · x.
1x = -6
5 ÷ 5 = 1, -30 ÷ 5 = -6
x = -6
Identity Property; 1x = x
To check your solution, replace x with -6 in the original equation. CHECK
5x = -30 5(-6) -30 -30 = -30
Write the equation. Check to see whether this statement is true. The statement is true.
The solution is -6. To graph it, draw a dot at -6 on a number line. ⫺7 ⫺6
⫺5
⫺4
⫺3
⫺2 ⫺1
1. Solve -48 = 6x. Check your solution and graph it on a number line.
PARKS It costs $3 per car to use the hiking trails along the Columbia River Highway. If income from the hiking trails totaled $1275 in one day, how many cars entered the park? The cost per car
Words
$3
Equation
The Columbia River Highway, built in 1913, is a historic route in Oregon that curves around twenty waterfalls through the Cascade Mountains. Source: columbiariverhighway.com
Write the equation.
3x 1275 _ =_
Divide each side by 3.
3
x = 425 CHECK
equals
the total
x
=
$1275
·
3x = 1275 3
the number of cars
Let = the number of cars.
Variable
Real-World Link
times
Simplify.
3x = 1275 3(425) 1275 1275 = 1275
Write the equation. Check to see whether this statement is true. The statement is true.
Therefore, 425 cars entered the park.
2. PARKS In-state camping permits for New Mexico State Parks cost $180 per year. If income from the camping permits totaled $8280 during the first day of sales, how many people bought permits? Personal Tutor at pre-alg.com
142 Chapter 3 Equations Stuart Westmorland/CORBIS
Solve Equations by Multiplying Some equations can be solved by multiplying each side by the same number. This property is called the Multiplication Property of Equality. Multiplication Property of Equality Interactive Lab pre-alg.com
Words
When you multiply each side of an equation by the same number, the two sides remain equal.
Symbols
For any numbers a, b, and c, if a = b, then ac = bc.
_x = 7
8=8
Examples
6
_6x 6 = (7)6
8(-2) = 8(-2)
x = 42
-16 = -16
EXAMPLE Reading Math Division Expressions y Remember, _ means -4 y divided by -4.
Solve Equations by Multiplying
y -4
Solve _ = -9. Check your solution and graph it on a number line. y _ = -9 -4 y _ (-4) = -9(-4) -4
y = 36
Write the equation. y -4
Multiply each side by −4 to undo the division in _. Simplify.
y CHECK _ = -9 -4
Write the equation.
36 _ -9
Check to see whether this statement is true.
-9 = -9
The statement is true.
-4
The solution is 36. To graph it, draw a dot at 36 on a number line. ÎÓ
ÎÎ
Î{
Îx
ÎÈ
ÎÇ
În
x 3. Solve 7 = _ . Check your solution and graph it on a number line. -2
Example 1 (p. 142)
Example 2 (p. 142)
Example 3 (p. 143)
ALGEBRA Solve each equation. Check your solution. 1. 4x = 24
2. -2a = 10
3. -7t = -42
4. TOYS A spiral toy that can bounce down a flight of stairs is made from 80 feet of wire. Write and solve an equation to find how many of these toys can be made from a spool of wire that contains 4000 feet. ALGEBRA Solve each equation. Check your solution. k =9 5. _ 3
Extra Examples at pre-alg.com
y 6. _ = -8 5
n 7. -11 = _ -6
Lesson 3-4 Solving Equations by Multiplying or Dividing
143
HOMEWORK
HELP
For See Exercises Examples 8–31 1, 3 32–35 2
ALGEBRA Solve each equation. Check your solution. 8. 3t = 21
h 10. _ =6
9. 8x = 72
4
11. _c = 4
g 12. _ = -7 -2
x 13. -42 = _
14. -32 = 4y
15. 5n = -95
16. -56 = -7p
17. -8j = -64
b 18. 11 = _
h 19. _ = 20
20. 45 = 5x
21. 3u = 51
m = -3 22. _
24. 86 = -2v
25. -8a = 144
v 27. _ = -132
k 28. -21 = _
30. -116 = -4w
31. -68 = -4m
9
d 23. _ = -3 3 f 26. _ = -10 -13 29. -56 = _t 9
-2
-3
-7 45
-11
8
32. BOATING A forest preserve rents canoes for $12 per hour. Corey has $36. Write and solve an equation to find how many hours he can rent a canoe.
Indian Ocean
Pacific Ocean Outback
AUSTRALIA Southern Ocean
Real-World Link Some students living in the Outback are so far from schools that they get their education by special radio programming. They mail in their homework and sometimes talk to teachers by two-way radio. Source: Kids Discover Australia
33. FRUIT Jenny picked a total of 960 strawberries in 1 hour. Write and solve an equation to find how many strawberries Jenny picked per minute. 34. RANCHING The largest ranch in the world is in the Australian Outback. It is about 12,000 square miles, which is five times the size of the largest United States ranch. Write and solve an equation to find the size of the largest United States ranch. 35. RANCHING In the driest part of an Outback ranch, each cow needs about 40 acres for grazing. Write and solve an equation to find how many cows can graze on 720 acres of land. ALGEBRA Graph the solution of each equation on a number line. 36. -6r = -18
37. -42 = -7x
n 38. _ =3 12
MEASUREMENT The chart shows several conversions in the customary system. Write and solve an equation to find each quantity. 40. the number of feet in 132 inches 41. the number of yards in 15 feet 42. the number of miles in 10,560 feet
y 39. _ = -1 -4
Customary System (length) 1 mile = 5280 feet 1 mile = 1760 yards 1 yard = 3 feet 1 foot = 12 inches 1 yard = 36 inches
EXTRA
PRACTICE
43. PAINTING A person-day is a unit of measure that represents one person working for one day. A painting contractor estimates that it will take 24 person-days to paint a house. Write and solve an equation to find how many painters the contractor will need to hire to paint the house in 6 days.
See pages 766, 796 Self-Check Quiz at pre-alg.com
44. FIND THE DATA Refer to the United States Data File on pages 18–21 of your book. Choose some data and write a real-world problem in which you would solve an equation by multiplying or dividing.
144 Chapter 3 Equations
H.O.T. Problems
45. OPEN ENDED Write an equation of the form ax = c where a and c are integers and the solution is 4. 46. NUMBER SENSE Find an equation that is equivalent to -9t = 18. x = 3, what is the value of 7x + 13? 47. CHALLENGE If 10
48.
Writing in Math Explain how equations are used to find the U.S. value of foreign currency. Illustrate your reasoning by finding the cost in U.S. dollars of a 12-pound bus trip in Egypt, if 6 pounds can be exchanged for one U.S. dollar.
49. Suppose that one pyramid balances two cubes and one cylinder balances three cubes as shown below. Which statement is NOT true?
50. The solution of which equation is NOT graphed on the number line below? { Î
A One pyramid and one cube balance three cubes. B One pyramid and one cube balance one cylinder.
Ó
£
ä
£
Ó
Î
F 12 = -6x
H -14 = 7x
G 8x = -16
J -18x = -36
51. During a vacation, the Mulligan family drove 63 miles in 1 hour. If they averaged the same speed during their trip, which equation can be used to find how far the Mulligan family drove in 6 hours? 63 A _ x =6
C One cylinder and one pyramid balance four cubes. D One cylinder and one cube balance two pyramids.
x = 63 B _ 6
C 6x = 63 D 63x = 6
ALGEBRA Solve each equation. Check your solution. (Lesson 3-3) 52. 3 + y = 16
53. 29 = n + 4
54. k - 12 = -40
ALGEBRA Simplify each expression. (Lesson 3-2) 55. 4x + 7x
56. 2y + 6 + 5y
57. 3 - 2(y + 4)
58. AGE Patricia is 12 years old, and her younger sister Renee is 2 years old. How old will each of them be when Patricia is twice as old as Renee? (Lesson 1-1)
PREREQUISITE SKILL Find each difference. (Lesson 2-3) 59. 8 - (-2)
60. -5 - 5
61. -10 - (-8)
62. -18 - 4
63. -45 - (-9)
64. 33 - (-19)
Lesson 3-4 Solving Equations by Multiplying or Dividing
145
CH
APTER
3
Mid-Chapter Quiz Lessons 3-1 through 3-4
1. MULTIPLE CHOICE Lucita works at a fitness center and earns $5.50 per hour. She worked 3 hours on Friday and 7 hours on Saturday. Which expression does NOT represent her wages that weekend? (Lesson 3-1) A 5.50(3 + 7) B 10(5.50)
9. AVIATION On December 17, 1903, the Wright brothers made the first flights in a power-driven airplane. Orville’s flight covered 120 feet, which was 732 feet shorter than Wilbur’s. Find the length of Wilbur Wright’s flight. (Lesson 3-3) 10. WEATHER Before a storm, the barometric pressure dropped to 29.2, which was 1.3 lower than the pressure earlier in the day. Write an equation to represent this situation.
C 5.50(3) + 5.50(7) D 7(5.50 + 3) 2. FUND-RAISING Debbie sold 23 teen magazines at $3.25 each, 38 sports magazines at $3.50 each, and 30 computer magazines at $2.95 each. How much money did Debbie raise? (Lesson 3-1) Simplify each expression. (Lessons 3-1 and 3-2) 3. 6(x + 2) 4. 5(x - 7) 5. 6y - 4 + y 6. 2a + 4(a - 9) 7. SCHOOL You spent m minutes studying on Monday. On Tuesday, you studied 15 more minutes than you did on Monday. Write an expression in simplest form that represents the total amount of time spent studying on Monday and Tuesday. (Lesson 3-2)
(Lesson 3-3)
ALGEBRA Solve each equation. (Lessons 3-3 and 3-4) 11. 4h = -52 12. y - 5 = -23 x =4 13. -3
14. n + 16 = 44 15. MULTIPLE CHOICE The table shows the five nearest train stops to Main Street. Which equation will best help you find how much further Peach Court is from Main Street than City Center is from Main Street? (Lesson 3-3)
Train Stop City Center 14th Street Grand Hotel Stadium Peach Court
8. MULTIPLE CHOICE A paving brick is shown. Find the perimeter of 5 bricks. (Lesson 3-2) ÎX ÓX Ó
Distance to Main Street (miles) 4 6 7 12 17
A x - 17 = 4 ÓX Ó
B x + 17 = 4 C x - 4 = 17
xX Î
D x + 4 = 17
F 12x + 1 G 40x + 10 H 60x + 5 J 50x - 10 146 Chapter 3 Equations
16. MONEY Ricardo spends $3.50 for lunch each day. Write and solve an equation to find how long it takes him to spend $21 on lunch. (Lesson 3-4)
3-5
Solving Two-Step Equations
Main Idea • Solve two-step equations.
The equation 2x + 1 = 9, modeled below, can be solved with algebra tiles.
New Vocabulary two-step equation
X
X £
£
£
£
£
£
£
£
£
£
ÓX £
Step 1 Remove 1 tile from each side of the mat.
X
X £
£
£
£
£
£
£
£
£
£
ÓX £ £
£
Step 2 Separate the remaining tiles into two equal groups. X
£
£
£
£
£
£
£
£
X
ÓX
n
a. What property is shown by removing a tile from each side? b. What property is shown by separating the tiles into two groups? c. What is the solution of 2x + 1 = 9?
Solve Two-Step Equations A two-step equation contains two operations. In the equation 2x + 1 = 9, x is multiplied by 2 and then 1 is added. To solve two-step equations, use inverse operations to undo each operation in reverse order. Step 1 First, undo addition. 2x + 1 = 9 2x + 1 - 1 = 9 - 1 Subtract 1 from each side. 2x = 8 Step 2 Then, undo multiplication. 2x = 8 8 2x _ =_ 2
2
Divide each side by 2.
x=4 The solution is 4. Lesson 3-5 Solving Two-Step Equations
147
EXAMPLE
Solve Two-Step Equations
a. Solve 5x - 2 = 13. Check your solution. 5x - 2 = 13
Write the equation.
5x - 2 + 2 = 13 + 2 Undo subtraction. Add 2 to each side. 5x = 15
Simplify.
5x 15 _ =_
Undo multiplication. Divide each side by 5.
5
5
x=3 CHECK
Simplify.
5x - 2 = 13
Write the equation.
5(3) - 2 13
Check to see whether this statement is true.
13 = 13
The statement is true.
The solution is 3. n b. Solve 4 = _ + 11.
4 4 - 11 -7 6(-7) -42
6 n = _ + 11 Write the equation. 6 n =_ + 11 - 11 Undo addition. Subtract 11 from each side. 6 n =_ Simplify. 6 n =6 _ Undo division. Multiply each side by 6. 6 =n Check your solution.
( )
Solve each equation. n 1B. _ + 15 = 8
1A. 6x + 1 = 25
3
SALES Liana bought a DVD recorder. If she pays $80 now, her monthly payments will be $32. The total cost will be $400. Solve 80 + 32x = 400 to find how many months she will make payments. 80 + 32x = 400 Checking Your Solution Use estimation to determine whether your solution is reasonable: 80 + 30(10) = 380. Since $380 is close to $400, the solution is reasonable.
Write the equation.
80 - 80 + 32x = 400 - 80 Subtract 80 from each side. 32x = 320
Simplify.
32x 320 _ =_
Divide each side by 32.
32
32
x = 10
Simplify.
Therefore, Liana will make payments for 10 months.
2. COMPUTERS Salvatore purchased a computer for $550. He paid $105 initially, and then he will pay $20 per month until the computer is paid off. Solve 105 + 20x = 545 to find how many payments he will make. 148 Chapter 3 Equations
EXAMPLE
Equations with Negative Coefficients
Solve 4 - x = 10. 4 - x = 10
Write the equation.
4 - 1x = 10
Identity Property; x = 1x
4 + (-1x) = 10
Definition of subtraction
-4 + 4 + (-1x) = -4 + 10 -1x = 6
Simplify.
6 -1x _ =_
Divide each side by -1.
-1
-1
x = -6
EXAMPLE
Check your solution.
Solve each equation. 3B. 35 - k = 21
3A. 19 = 9 - y
Mental Computation
Add -4 to each side.
Combine Like Terms Before Solving
Solve m - 5m + 3 = 47.
You use the Distributive Property to simplify 1m - 5m. 1m - 5m = (1 - 5)m = -4m You can also simplify the expression mentally.
m - 5m + 3 = 47 1m - 5m + 3 = 47 -4m + 3 = 47 -4m + 3 - 3 = 47 - 3
Write the equation. Identity Property; m = 1m Combine like terms, 1m and –5m. Subtract 3 from each side.
-4m = 44
Simplify.
-4m 44 _ =_
Divide each side by – 4.
-4
-4
m = -11
4A. 4 - 9d + 3d = 58
Simplify.
Solve each equation. 4B. 34 = 4m - 2 + 2m
Personal Tutor at pre-alg.com
Example 1 (p. 148)
Example 2 (p. 148)
Examples 3, 4 (p. 149)
ALGEBRA Solve each equation. Check your solution. 1. 2x - 7 = 9
2. -16 = 6a - 4
y 3. _ + 2 = 10 3
4. MEDICINE For Jillian’s cough, her doctor says that she should take eight tablets the first day and then four tablets each day until her prescription runs out. There are 36 tablets. Solve 8 + 4d = 36 to find how many more days she will take four tablets. ALGEBRA Solve each equation. Check your solution. 5. -7 - 8d = 17
6. 1 - 2k = -9
8. 2a - 8a = 24
9. -4 = 8y - 9y + 6
Extra Examples at pre-alg.com
-n -5 7. 8 = _ 7
10. -6j + 4 + 3j = -23
Lesson 3-5 Solving Two-Step Equations
149
HOMEWORK
HELP
For See Exercises Examples 11–28 1 29–30 2 31–34 3 35–42 4
ALGEBRA Solve each equation. Check your solution. 11. 3x + 1 = 7
12. 5x - 4 = 11
13. 4h + 6 = 22
14. 8n + 3 = -5
15. 37 = 4d + 5
16. 9 = 15 + 2p
17. 2n - 5 = 21
18. 3j - 9 = 12
19. -1 = 2r - 7
20. 12 = 5k - 8 23. 3 + _t = 35 2
w 26. _ - 4 = -7 8
y 21. 10 = 6 + _ 7 p 24. 13 + _ = -4 3 c 27. 8 = _ + 15 -3
n 22. 14 = 6 + _ 5
k 25. _ - 10 = 3 5
b 28. -42 = _ +8 -4
29. POOLS There were 640 gallons of water in a 1600-gallon pool. Water is being pumped into the pool at a rate of 320 gallons per hour. Solve 1600 = 320t + 640 to find how many hours it will take to fill the pool. 30. PHONE CARDS A telephone calling card allows for 25¢ per minute plus a one-time service charge of 75¢. If the total cost of the card is $5, solve 25m + 75 = 500 to find the number of minutes you can use the card. ALGEBRA Solve each equation. Check your solution. 31. 8 - t = -25
32. 3 - y = 13
33. 8 = -5 - b
34. 10 = -9 - x
35. 2w - 4w = -10
36. 3x - 5x = 22
37. x + 4x + 6 = 31
38. 5r + 3r - 6 = 10
39. 1 - 3y + y = 5
40. 16 = w - 2w + 9
41. 23 = 4t - 7 - t
42. -4 = -a + 8 - 2a
ALGEBRA Find each number. 43. Five more than twice a number is 27. Solve 2n + 5 = 27. n 44. Ten less than the quotient of a number and 2 is 5. Solve _ - 10 = 5. 2
45. Three less than four times a number is -7. Solve 4n - 3 = -7. n 46. Six more than the quotient of a number and 6 is -3. Solve _ + 6 = -3. 6
EXTRA
PRACTICE
See pages 766, 796. Self-Check Quiz at pre-alg.com
H.O.T. Problems
47. PERSONAL CARE In nine visits to the styling salon, Andre had spent $169 for haircuts. Of that amount, $16 was in tips. Write and solve an equation to find how much Andre pays for each haircut before the tip. 48. BUSINESS Jarret bought old bikes at an auction for $350. He fixed them and sold them for $50 each. He made a $6200 profit. Write and solve an equation to determine how many bikes he sold. 49. CHALLENGE The model represents the equation 6y + 1 = 3x + 1. What is the value of x? 50. OPEN ENDED Write a two-step equation that could be solved by using the Addition and Multiplication Properties of Equality. 51.
150 Chapter 3 Equations
Writing in Math Use the information about solving equations on page 147 to explain how algebra tiles can show the properties of equality. Illustrate your reasoning by showing how to solve 2x + 3 = 7 using algebra tiles.
52. GRIDDABLE The cost to park at an art fair is a flat rate plus a per-hour fee. The graph shows the cost for parking up to 4 hours. If x represents the number of hours and y represents the total cost, what is the cost in dollars for 7 hours? 4
53. A local health club charges an initial fee of $45 for the first month and then a $32 fee each month after that. The table shows the cost to join the health club for up to 6 months. What is the cost to join the health club for 10 months?
y
1 Cost (dollars) 45 Months
3 2 1 O
2 77
3 4 5 6 109 141 173 205
A $215
C $333
B $320
D $450
1 2 3 4 5 6 7 8x
ALGEBRA Solve each equation. Check your solution. (Lessons 3-3 and 3-4) 54. 5y = 60
55. 14 = -2n
x 56. _ = -9
57. x - 4 = -6
58. -13 = y + 5
59. 18 = 20 + x
3
ALGEBRA Simplify each expression. (Lesson 3-1) 60. 4(x + 1)
61. -5(y + 3)
62. 3(k - 10)
63. -9(y - 4)
64. 7(a - 2)
65. -8(r - 5)
66. ANALYZE TABLES The table shows the average game attendance for three football teams in consecutive years. What was the total change in attendance from Year 1 to Year 2 for the Bobcats? (Lesson 2-3)
Team
Year 1
Year 2
Bobcats Cheetahs Wildcats
6234 7008 6873
5890 7162 6516
Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7)
67. age and number of siblings
68. temperature and sales of sunscreen
PREREQUISITE SKILL Write an algebraic expression for each verbal expression. (Lesson 1-3)
69. two times a number less six 70. the quotient of a number and 15 71. the difference between twice a number and 8 72. three times a number increased by 10 73. the sum of 2x, 7x, and 4
Lesson 3-5 Solving Two-Step Equations
151
Translating Verbal Problems into Equations An important skill in algebra is translating verbal problems into equations. Consider the following situation. Jennifer is 6 years older than Akira. The sum of their ages is 20. You can explore this problem situation by asking and answering questions. Who is older? Jennifer How many years older? 6 If Akira is x years old, how old is Jennifer? x + 6 You can summarize this information in an equation.
Words
Jennifer is 6 years older than Akira. The sum of their ages is 20.
Variable
Let x = Akira’s age. Let x + 6 = Jennifer’s age.
Equation
x + ( x + 6) = 20
Exercises For each verbal problem, answer the related questions. 1. Lucas is 5 inches taller than Tamika, and the sum of their heights is 137 inches. a. Who is taller? b. How many inches taller? c . If x represents Tamika’s height, how tall is Lucas? d. What expression represents the sum of their heights? e. What equation represents the sentence the sum of their heights is 137? 2. There are five times as many students as teachers on the field trip, and the sum of students and teachers is 132. a. Are there more students or teachers? b. How many times more? c . If x represents the number of teachers, how many students are there? d. What expression represents the sum of students and teachers? e. What equation represents the sum of students and teachers is 132? 152 Chapter 3 Equations
3-6
Writing Two-Step Equations
Main Ideas Logan collected pledges for the charity walk-a-thon. He is receiving total contributions of $68 plus $20 for every mile that he walks. The table shows how to find the total amount that Logan could raise.
• Write verbal sentences as two-step equations. • Solve verbal problems by writing and solving two-step equations.
Number of Miles
Total Amount Raised
20(0) + 68 = $68
4
20(4) + 68 = $148
6
20(6) + 68 = $188
10
20(10) + 68 = $268
16
20(16) + 68 = $388
a. Write an expression that represents the amount Logan can raise when he walks m miles.
b. Suppose Logan raised $308. Write and solve an equation to find the number of miles Logan walked. c. Why is your equation considered to be a two-step equation?
Write Two-Step Equations In Chapter 1, you learned how to write Review Vocabulary Expression any combination of numbers and operations; Example: x - 3 (Lesson 1-2)
verbal phrases as expressions. Phrase
the sum of 20 times some number and 68
Expression
20n
+ 68
An equation is a statement that two expressions are equal. The expressions are joined with an equals sign. You can write verbal sentences as equations. Sentence
The sum of 20 times some number and 68 is 308.
Equation
EXAMPLE
20n + 68
= 308
Translate Sentences into Equations
Translate each sentence into an equation. Sentence a. Six more than twice a number is -20.
Equation 2n + 6 = -20
b. Eighteen is 6 less than four times a number.
18 = 4n - 6
c. The quotient of a number and 5, increased by 8, is equal to 14.
n _ + 8 = 14 5
1A. Four more than three times a number is -26. 1B. Twenty-four is 6 less than twice a number. 1C. The quotient of a number and 7, increased by 6, is equal to 12. Extra Examples at pre-alg.com
Lesson 3-6 Writing Two-Step Equations
153
EXAMPLE
Translate and Solve an Equation
Seven more than three times a number is 31. Find the number. Let n = the number.
Equations Look for the words is, equals, or is equal to when you translate sentences into equations.
3n + 7 = 31
Write the equation.
3n + 7 - 7 = 31 - 7 Subtract 7 from each side. 3n = 24 n=8
Simplify. Mentally divide each side by 3.
Therefore, the number is 8.
2. Translate the following sentence into an equation. Then find the number. Eight less than three times a number is -23.
Two-Step Verbal Problems In some real-world situations you start with a given amount and then increase it at a certain rate. These situations can be represented by two-step equations.
CELL PHONES Suppose you are saving money to buy a cell phone that costs $100. You have already saved $60 and plan to save $5 each week. How many weeks will you need to save? Explore
You have already saved $60. You plan to save $5 each week until you have $100.
Plan
Organize the data for the first few weeks in a table. Notice the pattern. Write an equation to represent the situation. Let x = the number of weeks. 5x + 60 = 100
Solve
5x + 60 = 100 5x + 60 - 60 = 100 - 60 5x = 40 x=8
Week
Amount
0 1 2 3
5(0) + 60 = 60 5(1) + 60 = 65 5(2) + 60 = 70 5(3) + 60 = 75
Write the equation. Subtract 60 from each side. Simplify. Mentally divide each side by 5.
You need to save $5 each week for 8 weeks. Real-World Link About 100 million cell phones in the United States are retired each year. Source: Inform Inc.
Check
If you save $5 each week for 8 weeks, you’ll have an additional $40. The answer appears to be reasonable.
3. SHOPPING Jasmine bought 6 CDs, all at the same price. The tax on her purchase was $7, and the total was $73. What was the price of each CD? Personal Tutor at pre-alg.com
154 Chapter 3 Equations Jim West/The Image Works
OLYMPICS In the 2004 Summer Olympics, the United States won 11 more medals than Russia. Together they won 195 medals. How many medals did the United States win? Let x = number of medals won by Russia. Then x + 11 = number of medals won by the United States.
Alternative Method Let x = number of U.S. medals. Then let x - 11 = number of Russian medals. x + (x - 11) = 195
x + (x + 11) = 195
Write the equation.
(x + x) + 11 = 195
Associative Property
2x + 11 = 195
Combine like terms.
2x + 11 - 11 = 195 - 11
Subtract 11 from each side.
x = 103
2x = 184
Simplify.
In this case, x is the number of U.S. medals, 103.
184 2x _ =_
Divide each side by 2.
2
2
x = 92
Simplify.
Since x represents the number of medals won by Russia, Russia won 92 medals. The United States won 92 + 11 or 103 medals.
4. CAR WASH During the spring car wash, the Activities Club washed 14 fewer cars than during the summer car wash. They washed a total of 96 cars during both car washes. How many cars did they wash during the spring?
Examples 1, 2 (pp. 153–154)
Translate each sentence into an equation. Then find each number. 1. Three more than four times a number is 23. 2. Four less than twice a number is -2. 3. The quotient of a number and 3, less 8, is 16. Solve each problem by writing and solving an equation.
Example 3 (p. 154)
Example 4 (p. 155)
HOMEWORK
HELP
For See Exercises Examples 6–11 1, 2 12, 13 3 14, 15 4
4. TEMPERATURE Suppose the current temperature is 17°F. It is expected to rise 3°F each hour for the next several hours. In how many hours will the temperature be 32°F? 5. AGES Lawana is five years older than her brother Cole. The sum of their ages is 37. How old is Lawana?
Translate each sentence into an equation. Then find each number. 6. Seven more than twice a number is 17. 7. Twenty more than three times a number is -4. 8. Four less than three times a number is 20. 9. Eight less than ten times a number is 82. 10. Ten more than the quotient of a number and -2 is three. 11. The quotient of a number and -4, less 8, is -42. Lesson 3-6 Writing Two-Step Equations
155
For Exercises 12–15, solve each problem by writing and solving an equation. 12. WILDLIFE Your friend bought 3 bags of wild birdseed and an $18 bird feeder. Each bag of birdseed costs the same amount. If your friend spent $45, find the cost of one bag of birdseed. 13. TEMPERATURE The temperature is 8°F. It is expected to fall 5° each hour for the next several hours. In how many hours will the temperature be -7°F? 14. POPULATION By 2020, California is expected to have 2 million more senior citizens than Florida, and the sum of the number of senior citizens in the two states is expected to be 12 million. Find the expected senior citizen population of Florida in 2020.
Real-World Career Meteorologist A meteorologist uses math to forecast the weather and analyzes how weather affects air pollution and agriculture.
15. BUILDINGS In New York City, the Chrysler Building is 320 feet taller than the Times Square Tower. The combined height of both buildings is 1772 feet. How tall is the Times Square Tower?
Building
Height (ft)
Citigroup Center
915
Chrysler Building
?
Empire State Building
1250
Times Square Tower
?
Woolworth Building
792
Source: emporis.com
Translate each sentence into an equation. Then find each number. 16. If 5 is decreased by 3 times a number, the result is -4.
For more information, go to pre-alg.com.
17. If 17 is decreased by twice a number, the result is 5. 18. Three times a number plus twice the number plus 1 is - 4. 19. Four times a number plus five more than three times the number is 47.
EXTRA
PRACTICE
See pages 767, 796. Self-Check Quiz at pre-alg.com
H.O.T. Problems
20. POPULATIONS Georgia’s Native-American population is 10,000 greater than Mississippi’s. Mississippi’s Native-American population is 106,000 less than Texas’. If the total population of all three is 149,000, find each state’s Native-American population. 21. CONSTRUCTION Henry is building a front door. The height of the door is 1 foot more than twice its width. If the door is 7 feet high, what is its width? 22. OPEN ENDED Write a two-step equation that has 6 as the solution. Write the equation using both words and symbols. 23. FIND THE ERROR Alicia and Ben are translating the following sentence into an equation: Three less than two times a number is 15. Who is correct? Explain your reasoning. Alicia 3 - 2x = 15
Ben 2x - 3 = 15
24. CHALLENGE If you begin with an even integer and count by two, you are counting consecutive even integers. Write and solve an equation to find two consecutive even integers whose sum is 50. 156 Chapter 3 Equations Dwayne Newton/PhotoEdit
25. NUMBER SENSE The table shows the expected population age 65 or older for certain states in 2030. Use the data to write a problem that can be solved by using a two-step equation. 26.
Population (age 65 or older) Number State (millions) CA 8.3
Writing in Math Explain how two-step equations are used to solve real-world problems. Formulate a problem situation that starts with a given amount and then increases.
FL
7.8
TX
5.2
NY
3.9
Source: U.S. Census Bureau
27. An electrician charges $35 for a house call and $80 per hour for each hour worked. If the total charge was $915 to wire a new house, which equation would you use to find the number of hours n that the electrician worked?
28. You and your friend spent a total of $15 for lunch. Your friend’s lunch cost $3 more than yours did. How much did you spend for lunch? F $6 G $7
A 35n + 2n(80) = 915
H $8
B 80 + 35n = 915
J $9
C 35 + (80 - n) = 915 D 35 + 80n = 915
ALGEBRA Solve each equation. Check your solution. (Lessons 3-3, 3-4, and 3-5) 29. 6 - 2x = 10
30. -4x = -16
31. y - 7 = -3
32. 7y + 3 = -11
33. CONCERTS A concert ticket costs t dollars, a hamburger costs h dollars, and soda costs s dollars. Write an expression that represents the total cost of a ticket, hamburger, and soda for n people. (Lesson 3-1) y
Name the ordered pair for each point graphed on the coordinate plane at the right. (Lesson 2-6) 34. T
35. C
36. R
T
R x
O
37. P
38. FOOD The SubShop had 36, 45, 41, and 38 customers during the lunch hour the last four days. Find the mean of the number of customers per day. (Lesson 2-5)
P
C
ALGEBRA Evaluate each expression if x = -12, y = 4, and z = -1. (Lesson 2-2) 39. ⎪x⎥ - 7
40.
⎪x⎥
+ ⎪y⎥
41. ⎪z⎥ - ⎪x⎥
42. ⎪y⎥ - ⎪x⎥ + ⎪z⎥
PREREQUISITE SKILL Find the next term in the pattern. (Lesson 1-1) 43. 5, 9, 13, 17, … 44. 326, 344, 362, 380, … 45. 20, 22, 26, 32, …
Lesson 3-6 Writing Two-Step Equations
157
3-7
Sequences and Equations
Main Ideas • Describe sequences using words and symbols. • Find terms of arithmetic sequences.
The table shows the distance a car moves during the time it takes to apply the brakes and while braking.
Speed (mph)
New Vocabulary
a. What is the braking distance for a car going 70 mph?
sequence arithmetic sequence term common difference
b. What is the difference in reaction distances for every 10-mph increase in speed?
Reaction Braking Distance (ft) Distance (ft)
20
20
20
30
30
45
40
40
80
50
50
125
60
60
180
c. Describe the braking distance as speed increases.
Describing Sequences A sequence is an ordered list of numbers. An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same. So, you can find the next term in the sequence by adding the same number to the previous term. Each number is called a term of the sequence.
EXAMPLE
20, +10
30,
40, +10
50,
+10
60, … The difference is called the common difference.
+10
Describe an Arithmetic Sequence
Describe the sequence 4, 8, 12, 16, … using words and symbols. +1
Substitute numbers from the table to check whether your equation is true.
158 Chapter 3 Equations
+1
Term Number (n)
1
2
3
4
Term (t )
4
8
12
16
+4
Check Your Answers
+1
+4
+4
The difference of the term numbers is 1. The terms have a common difference of 4. Also, a term is 4 times the term number. The equation t = 4n describes the sequence.
Describe each sequence using words and symbols. 1A. 10, 11, 12, 13, … 1B. 5, 10, 15, 20, …
Finding Terms Once you have described a sequence with a rule or equation, you can use the rule to extend the pattern and find other terms.
EXAMPLE
Find a Term in an Arithmetic Sequence
Find the 15th term of 7, 10, 13, 16, … . First write an equation that describes the sequence. +1
+1
+1
Term Number (n)
1
2
3
4
Term (t )
7
10
13
16
+3
+3
The difference of the term numbers is 1. The terms have a common difference of 3.
+3
The common difference is 3 times the difference in the term numbers. This suggests that t = 3n. However, you need to add 4 to get the exact value of t. Thus, t = 3n + 4. CHECK If n = 2, then t = 3(2) + 4 or 10. If n = 4, then t = 3(4) + 4 or 16. To find the 15th term in the sequence, let n = 15 and solve for t. t = 3n + 4 = 3(15) + 4 or 49
Write the equation.
So, the 15th term is 49.
2. Find the 20th term of 5, 8, 11, 14, … . Personal Tutor at pre-alg.com
Real-World Link The restaurant industry employs about 12.2 million people, making it the nation’s largest employer outside of government. Source: restaurant.org
RESTAURANTS The diagram shows the number of square tables needed to seat 4, 6, or 8 people at a restaurant. How many tables are needed to seat 16 people? Make a table to organize your sequence and find a rule. Number of Tables (t)
1
2
3
The difference of the term numbers is 1.
Number of People (p)
4
6
8
The terms have a common difference of 2.
The pattern in the table shows the equation p = 2t + 2. If p = 2t + 2 or 16 = 2t + 2, then t = 7. So, seven tables are needed to seat a party of 16.
3. RESTAURANTS Suppose the tables are shaped like hexagons. Find how many tables are needed for a group of 22 diners.
Extra Examples at pre-alg.com Don Tremain/Getty Images
Lesson 3-7 Sequences and Equations
159
Example 1 (p. 158)
Example 2 (p. 159)
Example 3 (p. 159)
Describe each sequence using words and symbols. 1. 2, 3, 4, 5, …
2. 6, 7, 8, 9, …
3. 3, 6, 9, 12, …
4. 7, 14, 21, 28, …
Write an equation that describes each sequence. Then find the indicated term. 5. 10, 11, 12, 13, …; 10th term
6. 6, 12, 18, 24, …; 11th term
7. 2, 5, 8, 11, …; 20th term
8. 2, 6, 10, 14, …; 14th term
9. GEOMETRY Suppose each side of a square has a length of 1 foot. Determine which figure will have a perimeter of 60 feet. &IGURE
HOMEWORK
HELP
For See Exercises Examples 10–21 1 22–29 2 30, 31 3
&IGURE
&IGURE
Describe each sequence using words and symbols. 10. 3, 4, 5, 6, …
11. 8, 9, 10, 11, …
12. 14, 15, 16, 17, …
13. 15, 16, 17, 18, …
14. 2, 4, 6, 8, …
15. 8, 16, 24, 32, …
16. 12, 24, 36, 48, …
17. 20, 40, 60, 80, …
18. 3, 5, 7, 9, …
19. 4, 6, 8, 10, …
20. 1, 4, 7, 10, …
21. 3, 7, 11, 15, …
Write an equation that describes each sequence. Then find the indicated term. 22. 16, 17, 18, 19, …; 23rd term
23. 14, 15, 16, 17, …; 16th term
24. 4, 8, 12, 16, …; 13th term
25. 11, 22, 33, 44, …; 25th term
26. 7, 10, 13, 16, …; 20th term
27. 7, 9, 11, 13, …; 33rd term
28. 1, 5, 9, 13, …; 89th term
29. 3, 8, 13, 18, …; 70th term
30. GEOMETRY Study the pattern. Which figure will have 40 squares? }ÕÀi £
}ÕÀi Ó
}ÕÀi Î
31. CONSTRUCTION A building frame consists of beams in the form of triangles. The frame of a new office building will use 27 beams. Use the pattern below to find the number of triangles that will be formed for the frame.
Î EXTRA
PRACTIICE
See pages 767, 796. Self-Check Quiz at pre-alg.com
x
Ç
THEATERS One section of a movie theater has 26 seats in the first row, 35 seats in the second row, 44 seats in the third row, and so on. 32. How many seats are in the eighth row? 33. If there are 10 rows of seats, how many seats are in the section?
160 Chapter 3 Equations
ANALYZE GRAPHS For Exercises 34 and 35, use the graph. 34. Write an equation for the points (x, y) graphed at the right. (Hint: Make a table of ordered pairs.)
Y
35. Find x when y is 101.
H.O.T. Problems
36. OPEN ENDED Write an arithmetic sequence whose common difference is -8.
X
"
37. CHALLENGE Use an arithmetic sequence to find the number of multiples of 6 between 41 and 523. 38.
Writing in Math
Explain how sequences can be used to make
predictions.
39. The expression 1 + 2n(n + 2) describes a pattern of numbers. If n represents a number’s position in the sequence, which pattern does the expression describe? A 7, 17, 31, 49, 71, . . . B 4, 7, 9, 17, 27, . . . C 7, 17, 27, 31, 49, . . .
40. Use the pattern Side Length in the table to 1 find the 2 equation that 3 shows the 4 relationship between the side 5 length s and perimeter p of a pentagon.
D 7, 9, 17, 27, 31, . . .
Perimeter
F p=5+s
H s = 5p + 5
G p = 5s
J s=5+p
5 10 15 20 25
ALGEBRA Translate each sentence into an equation. (Lesson 3-6) 41. Five more than three times a number is 20. 42. Thirty-six is 8 less than twice a number. 43. The quotient of a number and -10, less 3, is -63 ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 44. 6 - 3x = 21
45. 4y - 3 = 25
46. -3 + 2 z = -19
47. SOCCER A ticket to a soccer game is $12, a team pennant is $7, and a T-shirt is $15. Write two equivalent expressions for the total cost of a group outing for 10 people if each person buys a ticket, a pennant, and a T-shirt. Then find the cost. (Lesson 3-1) 48. WEATHER On Saturday, the temperature fell 10 degrees in 2 hours. Find the integer that expresses the temperature change per hour. (Lesson 2-5) PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 3-4) 49. 2x = -8
50. 15s = 75
51. 108 = 18x
52. 25z = 175
Lesson 3-7 Sequences and Equations
161
3-8
Using Formulas
Main Ideas • Solve problems by using formulas. • Solve problems involving the perimeters and areas of rectangles.
New Vocabulary formula perimeter area
Reading Math Formulas A formula is a concise way to describe a relationship among quantities.
The top recorded speed of a mallard duck in level flight is 65 miles per hour. You can make a table to record the distances that a mallard could fly at that rate.
Speed (mph)
Time (h)
Distance (mi)
65
1
65
b. What disadvantage is there in showing the data in a table?
65
2
130
65
3
195
c. Describe an easier way to summarize the relationship between the speed, time, and distance.
65
t
?
a. Write an expression for the distance traveled by a duck in t hours.
Formulas A formula is an equation that shows a relationship among certain quantities. A formula usually contains two or more variables. One of the most commonly used formulas is d = rt, which shows the relationship between distance d, rate (or speed) r, and time t.
SCIENCE What is the rate in miles per hour of a dolphin that travels 120 miles in 4 hours? Method 1 Substitute first. d = rt 120 = r · 4 120 = r · 4 4 4
30 = r
Method 2 Solve for r first.
Write the formula.
d = rt
Replace d with 120 and t with 4.
t = t
Divide each side by t.
Divide each side by 4.
d =r t
Simplify.
Simplify.
d
rt
120 = r 4
30 = r
Write the formula.
Replace d with 120 and t with 4. Simplify.
The dolphin travels at a rate of 30 miles per hour.
1. SCIENCE How long does it take a zebra to travel 160 miles at a speed of 40 miles per hour? 162 Chapter 3 Equations Getty Images
Perimeter and Area Formulas are commonly used in measurement. The distance around a geometric figure is called the perimeter. One method of finding the perimeter P of a rectangle is to add the measures of the four sides. Perimeter of a Rectangle Words
The perimeter of a rectangle is twice the sum of the length and width.
Symbols P = + + w + w P = 2 + 2w or P = 2( + w)
EXAMPLE Common Misconception Although the length of a rectangle is usually greater than the width, it does not matter which side you choose to be the length.
ᐉ
Model w
Find Perimeters and Lengths of Rectangles
a. Find the perimeter of the rectangle. P = 2( + w)
11 in.
Write the formula.
5 in.
= 2(11 + 5) Replace with 11 and w with 5. = 2(16)
Add 11 and 5.
= 32
Simplify. The perimeter is 32 inches.
b. The perimeter of a rectangle is 28 meters. Its width is 8 meters. Find the length. P = 2 + 2w
Write the formula.
28 = 2 + 2(8)
Replace P with 28 and w with 8.
28 = 2 + 16
Simplify.
28 - 16 = 2 + 16 - 16 12 = 2 6=
Subtract 16 from each side. Simplify. Mentally divide each side by 2.
The length is 6 meters.
2A. Find the perimeter of a rectangle with length 15 meters and width 10 meters. 2B. The perimeter of a rectangle is 26 yards. Its length is 8 yards. Find the width. The measure of the surface enclosed by a figure is its area. Area of a Rectangle Words
The area of a rectangle is the product of the length and width.
Symbols A = w
ᐉ
Model w
Extra Examples at pre-alg.com
Lesson 3-8 Using Formulas
163
EXAMPLE
Find Areas and Lengths of Rectangles
a. Find the area of a rectangle with length 15 meters and width 7 meters. 15 m A = w
Write the formula.
= 15 · 7
Replace with 15 and w with 7.
= 105
Simplify.
7m
The area is 105 square meters. b. The area of a rectangle is 45 square feet. Its length is 9 feet. Find its width. Method 1 Substitute, then solve for the variable.
Method 2 Solve, then substitute.
A = w Write the formula. 45 = 9w
Replace A with 45 and with 9.
5=w
Mentally divide each side by 9.
A = w
Write the formula.
w A _ =_
Divide each side by .
A _ =w
Simplify.
45 _ =w
Replace A with 45 and with 9.
9
5=w
Simplify.
The width is 5 feet.
3A. Find the area of a rectangle with a length of 14 inches and a width of 12 inches. 3B. The area of a rectangle is 198 square meters. Its width is 11 meters. Find its length. Personal Tutor at pre-alg.com
Example 1 (p. 162)
Examples 2, 3 (pp. 163–164)
1. ANIMALS How long would it take a bottlenose dolphin to swim 168 miles at 12 miles per hour? GEOMETRY Find the perimeter and area of each rectangle. 2.
3.
8 ft 3 ft
15 km 2 km
4. a rectangle with length 15 feet and width 6 feet GEOMETRY Find the missing dimension in each rectangle. 5.
6.
12 in.
8m
w Area 96 m Perimeter 32 in.
164 Chapter 3 Equations
ᐉ
2
HOMEWORK
HELP
For See Exercises Examples 7–8 1 9–25 2, 3
7. TRAVEL Find the distance traveled by driving at 55 miles per hour for 3 hours. 8. BALLOONING What is the rate, in miles per hour, of a balloon that travels 60 miles in 4 hours? GEOMETRY Find the perimeter and area of each rectangle. 9.
10.
3 mi
11.
9 cm
12 ft 5 ft
2 mi
12.
18 cm
13.
18 in.
14. 12 m 6m
50 in. 12 m
17 m
15. a rectangle that is 38 meters long and 10 meters wide 16. a square that is 5 meters on each side GEOMETRY Find the missing dimension in each rectangle. 17.
18.
15 cm Area 270 cm2
19.
w
Area 176 yd2
11 m 16 yd Perimeter 70 m
ᐉ
20.
Freddy Adu became the youngest professional player in modern American team sports history when he joined D.C. United at 14 years of age. Soccer is played on a rectangular field that is usually 120 yards long and 75 yards wide. Source: sportsillustrated. cnn.com
21.
7m
Real-World Link
ᐉ
w
w Area 154 in2
22.
12 ft
Area 468 ft2 ᐉ 14 in.
Perimeter 24 m
23. SOCCER Find the perimeter and area of the soccer field described at the left. 24. COMMUNITY SERVICE Each participant in a community garden is allotted a rectangular plot that measures 18 feet by 45 feet. How much fencing is needed to enclose each plot? 25. LANDSCAPING Jordan is paving a rectangular patio with bricks. If the patio contains a total of 308 bricks and there are 22 bricks running along the length of the patio, how many bricks run along the width of the patio? For Exercises 26 and 27, translate each sentence into a formula. 26. PROFITS The profit made during a year p is equal to sales s minus costs c. 27. GEOMETRY In a circle, the diameter d is twice the length of the radius r. Lesson 3-8 Using Formulas
Thanassis Stavrakis/AP/Wide World Photos
165
28. RUNNING The stride rate r of a runner is the number of strides n (or long steps) that he or she takes divided by the amount of time
Runner
Number of Strides
Time(s)
A B
20 30
5 10
n . The best runners usually have the t, or r = _ t
greatest stride rate. Which runner has the greater stride rate?
Using a formula can help you find the cost of a vacation. Visit pre-alg.com to continue work on your project.
GEOMETRY Draw and label the dimensions of each rectangle whose perimeter and area are given. 29. P = 14 ft, A = 12 ft2
30. P = 16 m, A = 12 m2
31. P = 16 cm, A = 16 cm2
LANDSCAPING For Exercises 32 and 33, use the figure at the right.
80 ft
32. What is the area of the lawn?
Lawn
33. Suppose your family wants to fertilize the lawn that is shown. If one bag of fertilizer covers 2500 square feet, how many bags of fertilizer should you buy?
House 28 ft x 50 ft
75 ft
Driveway 15 ft x 20 ft
34. TRAVEL An airplane flying at 500 miles per hour leaves Minneapolis. One-half hour later, a second airplane leaves Minneapolis flying in the same direction at a rate of 600 miles per hour. How long will it take the second airplane to overtake the first? GEOMETRY Find the area of each rectangle. 35.
Y
!
X
"
$
36.
"
Y %
X
"
# (
EXTRA
PRACTIICE
See pages 767, 795. Self-Check Quiz at pre-alg.com
H.O.T. Problems
&
'
BICYCLING For Exercises 37 and 38, use the following information. American Lance Armstrong won the 2005 Tour de France, completing the 2102-mile race in 83 hours, 36 minutes, 2 seconds. 37. Estimate Armstrong’s average rate in miles per hour for the race. 38. Armstrong also won the 2003 Tour de France. He completed the 2125-mile race in 80 hours, 2 minutes, 8 seconds. Without calculating, determine which race was completed with a faster average speed. Explain. 39. OPEN ENDED Draw and label a rectangle that has a perimeter of 18 inches. 40. CHALLENGE Is it sometimes, always, or never true that the perimeter of a rectangle is numerically greater than its area? Give an example to justify your answer.
166 Chapter 3 Equations
41. REASONING A rectangle has width w. Its length is one less than twice its width. Write an expression in simplest form for its perimeter. 42.
Writing in Math Explain why formulas are important in math and science. Include an example of a formula from math or science that you have used and an explanation of how you used the formula.
43. The formula d = rt can be rewritten as dt = r. How is the rate affected if the time t increases and the distance d remains the same?
44. The area of each square in the figure is 16 square units. Find the perimeter.
A It increases.
F 16 units
B It decreases.
G 32 units
C It remains the same.
H 48 units
D There is not enough information.
J 64 units
Write an equation that describes each sequence. Then find the indicated term. (Lesson 3-7) 45. 11, 12, 13, 14, …; 60th term
46. 4, 11, 18, 25, …; 100th term
47. Eight more than five times a number is 78. Find the number. (Lesson 3-6) LIGHT BULBS The table shows the average life of an incandescent bulb for selected years. (Lesson 1-6)
scen Light Bulbs Incandescent 1200 Hours
49. State the domain and the range of the relation.
1500
1500
48. Write a set of ordered pairs for the data.
1000
900
600
600 300 0
14
1870
1881 1910 Years
2000
Math and Technology I Need a Vacation! It’s time to complete your Internet project. Use the information and data you have
gathered about the costs of lodging, transportation, and entertainment for each of the vacations. Prepare a brochure or Web page to present your project. Be sure to include graphs and/or tables in the presentation. Cross-Curricular Project at pre-alg.com
Lesson 3-8 Using Formulas
167
EXTEND
Spreadsheet Lab
3-8
Perimeter and Area
A spreadsheet allows you to use formulas to investigate problems. When you change a numerical value in a cell, the spreadsheet recalculates the formula and automatically updates the results.
EXAMPLE Suppose a gardener wants to enclose a rectangular garden using part of a wall as one side and 20 feet of fencing for the other three sides. What are the dimensions of the largest garden she can enclose? If s represents the length of each side attached to the wall, 20 - 2s represents the length of the side opposite the wall. These values are listed in column B. The areas are listed in column C.
'ARDEN DIMENSIONSXLS "
!
,ENGTH OF 3IDE /PPOSITE 7ALL
3HEET
#
,ENGTH OF &ENCE ,ENGTH OF 3IDE !TTACHED TO 7ALL
3HEET
4HE SPREADSHEET EVALUATES THE FORMULA " !
!REA
4HE SPREADSHEET EVALUATES THE FORMULA ! "
3HEET
The greatest possible area is 50 square feet. It occurs when the length of each side attached to the wall is 5 feet, and the length of the side opposite the wall is 10 feet.
ANALYZE THE RESULTS 1. What is the area if the length of the side attached to the wall is 10 feet? 11 feet? 2. Are the answers to Exercise 1 reasonable? Justify your reasoning. 3. Suppose you want to find the greatest area that you can enclose with 30 feet of fencing. Which cell should you modify to solve this problem? 4. Use a spreadsheet to find the dimensions of the greatest area you can enclose with 40 feet, 50 feet, and 60 feet of fencing. 5. MAKE A CONJECTURE Use any pattern you may have observed in your answers to Exercise 4 to find the dimensions of the greatest area you can enclose with 100 feet of fencing. Explain. 168 Chapter 3 Equations
CH
APTER
3
Study Guide and Review
ownload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
>«ÌiÀ Î
µÕ>ÌÃ
Ûi *À«iÀÌÞ
£° ÃÌÀLÕÌ Ã } Ý«ÀiÃà Ӱ -«vÞ Î° µÕ>ÌÃ\ X {° µÕ>ÌÃ\
µÕ>ÌÃ x° /Ü-Ìi« Õ>ÌÃ È° 7ÀÌ} µ
Ç° -iµÕiVià n° ÀÕ>à ° ,iÛiÜ
Key Concepts Distributive Property
µÕ>ÌÃ
(Lesson 3-1)
• For any numbers a, b, and c, a(b + c) = ab + ac.
Solving Equations
formula (p. 162) inverse operations (p. 136) like terms (p. 129) perimeter (p. 163) sequence (p. 158) simplest form (p. 130) term (p. 129, 158) two-step equation (p. 147)
(Lessons 3-3 through 3-6)
• When you add or subtract the same number from each side of an equation, the two sides remain equal. • When you multiply or divide each side of an equation by the same nonzero number, the two sides remain equal. • To solve a two-step equation, undo operations in reverse order.
Sequences
area (p. 163) arithmetic sequence (p. 158) coefficient (p. 129) common difference (p. 158) constant (p. 129) Distributive Property (p. 124) equivalent equations (p. 136) equivalent expression (p. 124)
Vocabulary Check Complete each sentence with the correct term. Choose from the list above. 1. Terms that contain the same variables are called _________. 2. The ________ of a geometric figure is the measure of the distance around it. 3. In the term 4b, 4 is the ________ of the expression.
(Lesson 3-7)
• An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same.
4. The equations x + 3 = 8 and x = 5 are ________ because they have the same solution.
• A sequence can be described by a rule or equation that can be used to extend the pattern or find other terms in the pattern.
5. Addition and subtraction are ________ because they “undo” each other.
Perimeter and Area Formulas
6. In the expression 10x + 6, 6 is the ________ term.
(Lesson 3-8)
• The formula for the perimeter of a rectangle is P = 2( + w). • The formula for the area of a rectangle is A = w.
Vocabulary Review at pre-alg.com
7. The measure of the surface enclosed by a geometric figure is its ________. 8. A(n) ________ is an equation that shows a relationship among certain quantities.
Chapter 3 Study Guide and Review
169
CH
A PT ER
3
Study Guide and Review
Lesson-by-Lesson Review 3–1
The Distributive Property
(pp. 124–128)
Use the Distributive Property to write each expression as an equivalent algebraic expression. 9. 3(h + 6) 10. 7(x + 2) 11. -5(k + 1)
12. -2(a + 8)
13. (b - 4)(-2)
14. (y - 3)(-6)
Example 1 Use the Distributive Property to write 2(t - 3) as an equivalent algebraic expression. 2(t - 3) = 2[t + (-3)] Rewrite t - 3 as t + (-3).
15. BOWLING At the Bowling Palace, shoe rental is $3.00 and each game is $2.50. Write two equivalent expressions for the cost of a group of 3 people to rent shoes and play 2 games.
3–2
Simplifying Algebraic Expressions
Distributive Property
= 2t + (-6)
Simplify.
= 2t - 6
Definition of subtraction
(pp. 129–133)
Simplify each expression. 16. 4a + 5a 17. 3y + 7 + y
Example 2 Simplify 9x + 3 - 7x. 9x + 3 - 7x
18. x - 10 - 3x + 9
19. 4(m - 4) + 2
= 9x + 3 + (-7x) Definition of subtraction
20. 6w + 2(w + 9)
21. 8(n - 1) - 10n
= 9x + (-7x) + 3 Commutative Property
22. BASKETBALL Karen made 5 less than 4 times the number of free throws that Kimi made. Write an expression in simplest form that represents the total number of free throws made.
3–3
= 2t + 2(-3)
Solving Equations by Adding or Subtracting Solve each equation. Check your solution. 23. t + 5 = 8 24. 12 = b + 4 25. z - 10 = -6
26. a + 12 = -16
27. k - 1 = 4
28. -7 = n - 6
29. REPORTS Sonia needs to add 13 more pages to complete an assignment that is supposed to be 37 pages long. Write and solve an equation to find how many pages she has already completed. 170 Chapter 3 Equations
= [9 + (-7)]x + 3 Distributive Property = 2x + 3
Simplify.
(pp. 136–140)
Example 3 Solve x + 3 = 7. x+3=7
Write the equation.
x + 3 - 3 = 7 - 3 Subtract 3 from each side. x=4
Simplify.
Example 4 Solve y - 5 = -2. y - 5 = -2
Write the equation.
y - 5 + 5 = -2 + 5 Add 5 to each side. y=3
Simplify.
Mixed Problem Solving
For mixed problem-solving practice, see page 796.
3–4
Solving Equations by Multiplying or Dividing Solve each equation. Check your solution. 30. 6n = 48 31. -3x = 30 r = -2 32. -5
d 33. 22 =
-3
34. FASHION Rosa is making scarves for her friends. Each scarf requires 48 inches of material. Write and solve an equation to find how many scarves Rosa can make if she has 336 inches of material.
(pp. 141–145)
Example 5 Solve -5x = -30. -5x = -30
Write the equation.
-5x = -30 -5 -5
Divide each side by 5.
x=6
Simplify.
a = 3. Example 6 Solve -8 a =3 Write the equation. -8
a = -8(3) Multiply each side by -8. -8 -8
a = -24
3–5
Solving Two-Step Equations
(pp. 147–151)
Solve each equation. Check your solution. 35. 6 + 2y = 8 36. 3n - 5 = -17 37. _t + 4 = 2 3
38. _c - 3 = 2 9
Writing Two-Step Equations
Example 7 Solve 6k - 4 = 14. 6k - 4 = 14
Write the equation.
6k - 4 + 4 = 14 + 4
Add 4 to each side.
6k = 18
39. BOOKS Dion’s favorite book is 35 pages longer than Eva’s favorite book. The number of pages in both books is 271. Solve x + x + 35 = 271 to find the number of pages in Dion’s book.
3–6
Simplify.
k=3
Simplify. Divide each side by 6.
(pp. 154–157)
Translate each sentence into an equation. Then find each number. 40. Three more than twice a number is 53. 41. Six less than the quotient of a number and 4 is -3. 42. MONEY Suppose you are saving money to buy a digital video camera that costs $340. You have already saved $120 and plan to save $20 each week. How many weeks will you need to save?
Example 8 Translate the sentence into an equation. Then find the number. Seven less than three times a number is -22. 3n - 7 = -22
Write the equation.
3n - 7 + 7 = -22 + 7 Add 7 to each side. 3n = -15 n = -5
Simplify. Divide each side by 3.
Chapter 3 Study Guide and Review
171
CH
A PT ER
3 3–7
Study Guide and Review
Sequences and Equations
(pp. 158–161)
Describe each sequence using words and symbols. 43. 5, 6, 7, 8, … 44. 14, 15, 16, 17, … 45. 6, 12, 18, 24, …
46. 10, 20, 30, 40, …
Write an equation that describes each sequence. Then find the indicated term. 47. 8, 9, 10, 11, …; 19th term 48. 6, 10, 14, 18, …; 47th term 49. 7, 14, 21, 28, …; 70th term 50. GEOMETRY Which figure in the pattern below will have 99 squares?
3–8
}ÕÀi £
}ÕÀi Ó
Using Formulas
(pp. 162–167)
Example 9 Find the 35th term of 9, 18, 27, 36, … . Term Number (n)
1
2
3
4
Term (t)
9
18
27
36
The common difference is 9. Each term is 9 times the term number. So, t = 9n. t = 9n
Write the equation.
t = 9(35) Replace n with 35. t = 315
Simplify.
The 35th term of the sequence is 315.
}ÕÀi Î
Find the perimeter and area of each rectangle. 51.
Example 10 Find the perimeter and area of a 14-meter by 6-meter rectangle. P = 2( + w) Formula for perimeter = 2(14 + 6) Replace with 14 and w with 6. = 40 m
°
°
52. a rectangle with length 8 feet and width 9 feet 53. a square with sides that are 2.5 yards 54. SWIMMING Karl’s rectangular pool is 15 feet by 9.5 feet. What are the perimeter and area of the pool? 55. TYPING Toya typed a 1140-word essay in 12 minutes. At what rate is she typing? (Hint: Typing is measured in words per minute.)
172 Chapter 3 Equations
A=·w = 14 · 6 = 84
m2
Simplify. Formula for area Replace with 14 and w with 6. Simplify.
Example 11 The speed of light is 299,792,458 meters per second. Use the formula d = rt to calculate how far light travels in one minute. d=r·t
Write the formula.
= 299,792,458 · 60
Substitute.
= 17,987,547,480
Simplify.
CH
A PT ER
3
Practice Test
ENTERTAINMENT Suppose you pay $15 per hour to go horseback riding. You ride 2 hours today and plan to ride 4 more hours this weekend. 1. Write two different expressions to find the total cost of horseback riding. 2. Find the total cost. Simplify each expression. 3. x + 3x
4. 9x + 5 - x + 3
5. 10(y + 3) - 4y
6. -7b - 5(b - 4)
22. MULTIPLE CHOICE A carpet store advertises 16 square yards of carpeting for $300, which includes the $60 installation charge. Which equation could be used to determine the cost of one square yard of carpet x? A 16x = 300
C 60x + 16 = 300
B x + 60 = 300
D 16x + 60 = 300
23. MULTIPLE CHOICE In the sequence below, which expression can be used to find the value of the term in the nth position?
7. MUSIC Omar and Deb each have a digital music player. Deb has 37 more songs on her player than Omar has on his player. Write an expression in simplest form that represents the total number of songs on both players.
Position
Value
1
5
2
14
3
23
4
32
n
?
Solve each equation. Check your solution. 8. 19 = f + 5
9. -15 + z = 3
10. x - 7 = 16
11. g - 9 = -10
12. -8y = 72
n = -6 13. -30
14. 25 = 2d - 9
15. 4w - 18 = -34
16. 6v + 10 = -62
d +1 17. -7 =
18. x + 7 - 2x = 18
19. b - 7b + 6 = -30
F 5n
H 9n
G 5n + 4
J 9n - 4
24. Find the perimeter and area of the rectangle. 48 m 20 m
-5
20. TRAVEL Ms. Carter is renting a car from an agency that charges $20 per day plus $0.15 per mile. She has a budget of $80 per day. Use the equation 80 = 20 + 0.15m to find the maximum number of miles she can drive each day. 21. GENETICS Approximately one-seventh of the people in the world are left-handed. Write and solve an equation to estimate how many people in the United States are left-handed if the population of the United States is about 300 million.
Chapter Test at pre-alg.com
25. MULTIPLE CHOICE The rectangle below has a length of 20 centimeters and a perimeter of P centimeters. Which equation could be used to find the width of the rectangle?
Óä V
w A P = 40 + 2
B P = 40 + 2w C P = 20 + w D P = 20 + 2w
Chapter 3 Practice Test
173
CH
A PT ER
3
Standardized Test Practice Cumulative, Chapters 1–3
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. Which expression can be used to find the nth term of the following sequence, where n represents a number’s position in the sequence?
4. Let n represent the position of a number in the following sequence. 3
5 3
1 , 1 , , 1, , , 7 , 2, . . . 4 2 4 4 2 4
Which expression can be used to find any term in the sequence? n F 4
Position in Sequence
1
2
3
4
n
n G 2
Term
4
6
8
10
?
H 2n J 3n
A 2n B 2n + 1
5. A movie theater sells large boxes of popcorn for $7.50, medium boxes of popcorn for $5.75, and small boxes of popcorn for $3.75. Suppose a group of friends orders 2 large popcorns, 1 medium popcorn, and 2 small popcorns. Which equation can be used to find the total cost of the popcorn?
C 2n + 2 D 3n + 1
2. Mrs. Kelly’s deck has an area of 660 square feet.
A 2(7.5) + 5.75 + 2(3.75) B (2 + 1 + 2)(7.5 + 5.75 + 3.75)
FT
FT
7.5 + 5.75 + 3.75 C (2 + 1 + 2) 3
D 2(7.5) + 5.75 + 3.75
What is the length of the deck if the width is 22 feet? F 25 ft
H 32 ft
G 30 ft
J 35 ft
3. On Saturday the low temperature in Detroit, Michigan, was -7°F, and the high temperature was 23°F. How much warmer was the high temperature than the low temperature? A -30°F B -16°F
6. GRIDDABLE The ordered pairs (-7, -2), (-3, 5), and (-3, -2) are coordinates of three vertices of a rectangle. What is the y-coordinate of the ordered pair that represents the fourth vertex? 7. Todd is 5 inches taller than his brother. The sum of their heights is 139 inches. Find Todd’s height. F 67 in.
H 77 in.
G 72 in.
J 82 in.
8. Which expression represents the greatest integer?
C 16°F
A ⎪4⎥
C ⎪-8⎥
D 30°F
B ⎪-3⎥
D -9
174 Chapter 3 Equations
Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
9. The scatter plot shows how many cups of hot chocolate are sold on average at a ski resort based on the outside temperature.
12. GRIDDABLE Six tables positioned in a row will be used to display science projects. Each table is 8 feet long. How many yards of fabric are needed to make a banner that will extend from one end of the row of tables to the other?
Ì V>Ìi ->iÃ
Õ«Ã -` *iÀ ÕÀ
xx xä {x {ä Îx Îä Óx Óä £x £ä x
Y
ä
x yd
Óä Óx Îä Îx {ä {x "ÕÌÃ`i /i«iÀ>ÌÕÀi ®
13. Pedro is buying a DVD player that is on sale for $49.89 plus tax. If he pays with a $100 bill, what other information is needed to determine how much change he should receive?
X
A The brand of the DVD player
Which description best represents the relationship of the data? F Negative trend
B The amount of money Pedro has in his wallet
G No trend
C The amount of Pedro’s weekly income
H Positive trend
D The sales tax rate
J Cannot be determined Pre-AP 10. Henry paid $32 to rent a table saw for a two-day period. The maximum length of time a customer can rent the saw is 10 days. Which equation can be used to find c, the cost of renting the table saw for the maximum number of days? A c = 32 10
C c = 32 5
B c = 2 32
D c = 32 15
Record your answers on a sheet of paper. Show your work. 14. A basketball team can score 3-point baskets, 2-point baskets, and 1-point free throws. Josh heard the Springdale Stars scored a total of 63 points in their last game. Soledad says that they made a total of two 3-point baskets and 12 free throws in that game. a. Write an equation to represent the total points scored p. Use f for the number of free throws, g for the number of 2-point baskets, and h for the number of 3-point baskets. b. Can both Josh and Soledad be correct? Explain your reasoning.
11. Find - 19 - (-10). F -29
G -9
H9
J 29
Question 11 When a question involves integers, check to be sure that you have chosen an answer with the correct sign.
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Go to Lesson...
3-7
3-8
2-3
3-7
1-1
2-6
3-3
2-1
1-7
3-4
2-3
1-1
1-1
3-6
Chapter 3 Standardized Test Practice
175
Algebra and Rational Numbers Focus Understand rational numbers, ways of representing numbers, relationships among numbers, and number systems. Find measures of central tendency in data sets.
CHAPTER 4 Factors and Fractions Understand that different forms of numbers are appropriate for different situations. Select and use appropriate operations to solve problems and justify solutions.
CHAPTER 5 Rational Numbers Understand that different forms of numbers are appropriate for different situations. Use statistical procedures to describe data sets. Evaluate predictions and conclusions based on statistical data.
176 Unit 2 Algebra and Rational Numbers Envision/Corbis
Algebra and Nutrition You Are What You Eat Did you know that Wisconsin produces more cheese and grows more cranberries for processing than any other state? Calcium, found in cheese and cranberries, helps to keep your bones and teeth strong over your lifetime. In this project, you will be exploring how rational numbers are related to nutrition. Log on to pre-alg.com to begin.
Unit 2 Algebra and Rational Numbers
177
4
Factors and Fractions
•
Understand that different forms of numbers are appropriate for different situations
•
Select and use appropriate operations to solve problems and justify solutions
Key Vocabulary exponent (p. 180) power (p. 180) factor (p. 180) scientific notation (p. 214)
Real-World Link Technology The number of transistors on a computer chip (or the processing power of a computer) doubles about every 2 years.
actors and Fractions Make this Foldable to help you organize your notes about factors and fractions. egin with four sheets of notebook paper.
1 Fold four sheets of notebook paper in half from top to bottom.
3 Cut tabs into the margin. Make the top tab 2 lines wide, the next tab 4 lines wide, and so on.
178 Chapter 4 Factors and Fractions Mike Agliolo/Photo Researchers
2 Cut along the fold. Staple eight halfsheets together to form a booklet.
4 Label each of the tabs with the lesson number and title.
&ACTORS AN D &RACTIONS W BEL
GET READY for Chapter 4 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Evaluate each expression if x = 2, y = 5, and z = -1. (Lesson 1-3) 1. x + 12 2. z + (-5) 3. 4y + 8
4. 10 + 3z
5. (2 + y)9
6. 6(x - 4)
7. 3xy
8. 2z + y
9. POPULATION The freshman class has 15 more than twice the number of students as the previous year. If the freshman class had 140 students the previous year, how many freshman are there this year?
Example 1
Evaluate the expression 5x - 2y + 0z if x = -4, y = 9, and z = 6. 5x - 2y + 0z
Write the expression.
= 5(-4) - 2(9) + 0(6) Replace x with -4, y with 9, and z with 6.
= -20 - 18 + 0
Multiply.
= -38
Simplify.
(Lesson 1-3)
Example 2
Simplify. (Lesson 3-1) 10. 2(x + 1)
11. 3(n - 1)
Simplify -(x - 2).
12. -2(k + 8)
13. -4(x - 5)
-(x - 2) = -1(x - 2)
14. 6(2c + 4)
15. 5(-3s + t)
16. 7(a + b)
17. 9(b - 2c)
18. FOOD The concession stand offers a slice of pizza for $3.50 and a bottle of water for $1.25. Write two equivalent expressions for the total cost of p people each buying a slice of pizza and a bottle of water.
A negative sign before a parenthesis implies -1.
= (-1)(x) + (-1)(-2) Distribute -1 to each term inside the parentheses.
= -x + 2
Simplify.
(Lesson 3-1)
Find each product. (Prerequisite Skills, pp. 747–748) 19. 4.5 · 10 20. 3.26 · 100 21. 0.1 · 780
22. 15 · 0.01
23. 0.01 · 0.5
24. 301.8 · 0.001
25. HOTELS A hotel costs $159 plus 10% in taxes and fees for each night. The amount of taxes and fees is found by multiplying the cost of the hotel by 10% or 0.1. What is the cost of taxes and fees for a one-night stay? (Prerequisite Skills, pp. 747–748)
Example 3
Find the product of 0.1 × 2.78. 2.78 ×
← 2 decimal places
0.1 ← 1 decimal place
0.278
← 3 decimal places
The product is 0.278.
Chapter 4 Get Ready for Chapter 4
179
4-1
Powers and Exponents
Main Ideas • Write expressions using exponents. • Evaluate expressions containing exponents.
Computer memory is measured in small units called bytes. These units are based on products and factors of 2.
New Vocabulary factor base exponent power
a. Write 16 as a product of factors of 2. How many factors are there?
500MHZ
128MG CD-ROM
Memor y + Speed
Year
Amount of Memory in a Personal Computer
1980 1983 1992 1998 1999 2002 2004
16 kilobytes 1 megabyte 16 megabytes 32 megabytes 128 megabytes 512 megabytes 1 gigabyte
PC Sale 12/16
750MHZ 20 0GB
32MB + 32MB
40X Max.Var CDROM
1 Call 2 Write
b. How many factors of 2 form the product 128? c. One megabyte is Source: islandnet.com 1024 kilobytes. How many factors of 2 form the product 1024?
Exponents Two or more numbers that are multiplied to form a product are called factors. An expression like 2 × 2 × 2 × 2 with equal factors can be written as a power. A power has two parts, a base and an exponent. The expression 2 × 2 × 2 × 2 can be written as 24. The base is the number that is multiplied.
The exponent tells how many times the base is used as a factor.
24
The number that can be expressed using an exponent is called a power.
The table below shows how to write and read powers with positive exponents. Powers
Reading Math First Power When a number is raised to the first power, the exponent is usually omitted. So 21 is written as 2.
21
Words
Repeated Factors
23 24
2 to the first power 2 to the second power or 2 squared 2 to the third power or 2 cubed 2 to the fourth power or 2 to the fourth
2n
2 to the nth power or 2 to the nth
22
2 2·2 2·2·2 2·2·2·2 2·2·2·…·2 n factors
Any number, except 0, raised to the zero power is defined to be 1. 10 = 1 180 Chapter 4 Factors and Fractions
20 = 1
30 = 1
40 = 1
50 = 1
x0 = 1, x ≠ 0
EXAMPLE
Write Expressions Using Exponents
Write each expression using exponents. a. 3 · 3 · 3 · 3 · 3
b. t · t · t · t
The base is 3. It is a factor 5 times, so the exponent is 5. 3 · 3 · 3 · 3 · 3 = 35 Common Misconception (-9)2
is not the same as -92. (-9)2 = (-9)(-9) = 81 -92 = -1 · 92 = -81
The base is t. It is a factor 4 times, so the exponent is 4. t · t · t · t = t4
c. (-9)(-9)
d. (x + 1)(x + 1)(x + 1)
The base is -9. It is a factor 2 times, so the exponent is 2. (-9)(-9) = (-9)2
The base is x + 1. It is a factor 3 times, so the exponent is 3. (x + 1)(x + 1)(x + 1) = (x + 1)3
e. 7 · a · a · a · b · b First, group the factors with like bases. Then, write using exponents. 7 · a · a · a · b · b = 7 · (a · a · a) · (b · b) = 7a3b2 a · a · a = a3 and b · b = b2
Write each expression using exponents. 1A. 6 · 6 · 6 · 6 1B. x · x · x · x · x 1D. (c - d)(c - d) 1E. 9 · f · f · f · f · g
1C. (-2)(-2)(-2) 1F. (m + 1)
Evaluate Expressions Since powers are forms of multiplication, they need to be included in the rules for order of operations. Order of Operations Words
Example
Step 1
Simplify the expressions inside grouping symbols first.
Step 2
Evaluate all powers.
Step 3
Do all multiplications or divisions in order from left to right.
Step 4
Do all additions or subtractions in order from left to right.
(3 + 4)2 + 5 · 2 = 72 + 5 · 2 = 49 + 5 · 2 = 49 + 10 = 59
Reading Math Exponents An exponent goes with the number, variable, or quantity in parentheses immediately preceding it. • In 5 · 32, 3 is squared. 5 · 32 = 5 · 3 · 3 • In (5 · 3)2, (5 · 3) is squared. (5 · 3)2 = (5 · 3)(5 · 3)
Follow the order of operations to evaluate algebraic expressions.
EXAMPLE
Evaluate Numeric Expressions
Evaluate each expression. a. 23
b. 4 · 32
23 = 2 · 2 · 2 =8
2A. 54 Extra Examples at pre-alg.com
4 · 32 = 4 · 3 · 3 3 is a factor 2 times.
2 is a factor 3 times.
= 36
Multiply.
Multiply.
2B. 5 · 24 Lesson 4-1 Powers and Exponents
181
EXAMPLE
Evaluate Algebraic Expressions
Evaluate each expression. a. y2 + 5 if y = -3 Replace y with -3. y2 + 5 = (-3)2 + 5 = (-3)(-3) + 5 -3 is a factor two times. = 9 + 5 or 14 Multiply. Then add.
Powers of Negatives Be sure to follow the order of operations when evaluating powers. (-3)2 = -3 · (-3) or 9 and -32 = -(3 · 3) or -9.
b. 3(x + y)3 if x = -2 and y = 1 3(x + y)3 = 3(-2 + 1)3 Replace x with -2 and y with 1. = 3(-1)3 Simplify the expression inside the parentheses. = 3(-1) or -3 Evaluate (-1)3. Then simplify.
Evaluate each expression if a = 5 and b = -2. 3A. 10 + b2 3B. (a + b)3 Personal Tutor at pre-alg.com
Example 1 (p. 181)
Example 2 (p. 181)
Example 3 (p.182)
Write each expression using exponents. 1. n · n · n 4. 3 · 3 · x · x · x · x
2. 7 · 7 5. (y - 3)(y - 3)(y - 3)
3. (-4)(-4)(-4) 6. (a + 1)(a + 1)
8. 63
9. 2 · 52
Evaluate each expression. 7. 24
ALGEBRA Evaluate each expression if x = -2 and y = 4. 10. x3 - 4
11. 5(y - 1)2
12. x2 + y2
13. SOUND Fireworks can easily reach a sound of 169 decibels, which can be dangerous if prolonged. Write this number using a power greater than 1 and a lesser base.
HOMEWORK
HELP
For See Exercises Examples 14–25 1 26–34 2 35–46 3
Write each expression using exponents. 14. 17. 20. 23.
4·4·4·4·4·4 (-8)(-8)(-8)(-8) r·r·r·r 2·x·x·y·y
15. 18. 21. 24.
6 k·k m·m·m·m (n - 5)(n - 5)(n - 5)
16. 19. 22. 25.
(-5)(-5)(-5) (-t)(-t)(-t) a·a·b·b·b·b 9 · (p + 1) · (p + 1)
Evaluate each expression. 26. 72 29. (-2)5 32. 63 · 4 182 Chapter 4 Factors and Fractions
27. 103 30. 3 · 42 33. 35 · 10
28. (-9)3 31. 2 · 43 34. 20 · 10
ALGEBRA Evaluate each expression if a = 2, b = 4, and c = -3. 35. b4
36. c4
37. 4a4
38. ac3
39. b0 - 10
40. c2 + a2
41. 3a + b3
42. a2 + 3a - 1
43. b2 - 2b + 6
44. 3(b - 1)4
45. 2(3c + 7)2
46. 5(a3 + 6)
47. BIOLOGY A man burns approximately 121 Calories by standing for an hour. A woman burns approximately 100 Calories per hour when standing. Write each of these numbers as a power with an exponent greater than 1. 48. MILEAGE Which numbers in the table can be expressed as a power greater than 1? Name the cities and express the numbers as powers.
Miles to Kentucky Dam City
49. Write 7 cubed times x squared as repeated multiplication.
Miles
Bowling Green
120
Chicago
400
Evansville
100
Lexington
250
50. Write negative eight cubed using exponents and as a product of repeated factors.
Louisville
200
Nashville
125
51. Without using a calculator, order 96, 962, 9610, 965, and 960 from least to greatest. Explain.
Paducah
25
St. Louis
225
Source: kentuckylake.com
52. NUMBER THEORY Explain whether the square of any nonzero number is sometimes, always, or never a positive number.
Real-World Link The so noodles are about a yard long and as thin as a piece of yarn. Very few chefs still know how to make these noodles. Source: The Mathematics Teacher
FOOD For Exercises 53–55, use the following information. In an ancient Chinese tradition, a chef stretches and folds dough to make long, thin noodles called so. After the first fold, he makes 2 noodles. He stretches and folds it a second time to make 4 noodles. Each time he repeats this process, the number of noodles doubles. 53. Use exponents to express the number of noodles after each of the first five folds. 54. Legendary chefs have completed as many as thirteen folds. How many noodles is this? 55. If the noodles are laid end to end and each noodle is 5 feet long, after how many of these folds will the length be more than a mile? Replace each 56. 37
73
with , or = to make a true statement. 57. 24
42
58. 63
GEOMETRY For Exercises 59–61, use the cube at the right. 59. The surface area of a cube is the sum of the areas of the faces. Use exponents to write an expression for the surface area of the cube. EXTRA
I E PRACTIC
See pages 768, 797. Self-Check Quiz at pre-alg.com
60. The volume of a cube, or the amount of space that it occupies, is the product of the length, width, and height. Use exponents to write an expression for the volume of the cube.
44
3 cm
3 cm
61. If you double the length of each edge of the cube, are the surface area and volume also doubled? Explain. Lesson 4-1 Powers and Exponents
Christophe Loviny/CORBIS
3 cm
183
H.O.T. Problems
62. OPEN ENDED Use exponents to write a numerical expression and an algebraic expression in which the base is a factor 5 times. CHALLENGE Suppose the length of a side of a square is n units and the length of an edge of a cube is n units. 63. If all the side lengths of a square are doubled, are the perimeter and the area of the square doubled? Explain. 64. If all the side lengths of a square are tripled, show that the area of the new square is 9 times the area of the original square. Explain. 65. If all the edge lengths of a cube are tripled, show that the volume of the new cube is 27 times the volume of the original cube. Explain.
N
N
66. SELECT A TOOL Mercury has a mean distance from the Sun of 60002 miles. Which of the following tools might you use to determine the mean distance from the Sun to Mars if it is four times Mercury’s distance to the Sun? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model 67.
paper/pencil
calculator
Writing in Math Use the information about exponents on page 180 to explain how they are used to describe computer memory. Include an advantage of using exponents.
68. Which expression represents the number of cells after half an hour? A 210
C 220
B 215
D 230
Time (min)
Number of Bacteria
0 3 6 9 12
20 21 22 23 24
69. GRIDDABLE Suppose a certain forest fire doubles in size every 8 hours. If the initial size of the fire was 1 acre, how many acres will the fire cover in 3 days?
70. TORNADOES A tornado travels 300 miles in 2 hours. Use the formula d = rt to find the tornado’s speed in miles per hour. (Lesson 3-8) 71. PATTERNS Study the pattern. Find the equation that represents a relationship between the number of columns c and the number of rows r in the pattern. (Lesson 3-7) ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 72. 2x + 1 = 7
n 74. _ +8=6
73. 16 = 5k - 4
3
PREREQUISITE SKILL List all the factors for each number. (Pages 740–741) 75. 11
76. 10
184 Chapter 4 Factors and Fractions
77. 16
78. 50
EXTEND
4-1
Algebra Lab
Base 2
A computer contains a large number of tiny electronic switches that can be turned ON or OFF. The digits 0 and 1, also called bits, are the alphabet of computer language. This binary language uses a base two system of numbers.
The digit 1 represents the ON switch.
24
23
22
21
20
Place values are powers of 2.
1
1
1
The digit 0 represents the OFF switch.
101102 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (0 × 20) = 16
+
+
4
+
2
+
or 22
So, 101102 = 2210 or 22. You can also reverse the process and express base ten numbers as equivalent numbers in base two.
ACTIVITY Express the decimal number 13 as a number in base two. Step 1 Make a base 2 place-value chart. Find the greatest power of 2 that is less than 13. Place a 1 in that place value.
16
1 8
4
2
1
Step 2 Subtract 13 - 8 = 5. Now find the greatest power of 2 that is less than 5. Place a 1 in that place value.
16
1 8
1 4
2
1
16
1 8
1 4
0 2
1 1
Step 3 Subtract 5 - 4 = 1. Place a 1 in that place value. Step 4 There are no powers of 2 left, so place a 0 in any unfilled spaces.
So, 13 in the base 10 system is equal to 1101 in the base 2 system. Or, 13 = 11012.
ANALYZE THE RESULTS 1. Express 10112 as an equivalent number in base 10. Express each base 10 number as an equivalent number in base 2. 2. 6
3. 9
4. 15
5. 21
6. The first five place values for base 5 are shown. Any digit from 0 to 4 can be used to write a base 5 number. Write 179 in base 5.
625 125 25
5
1
7. OPEN ENDED Write 314 as an equivalent number in a base other than 2, 5, or 10. Include a place-value chart. 8. OPEN ENDED Choose a base 10 number and write it as an equivalent number in base 8. Include a place-value chart. Extend 4-1 Algebra Lab: Base 2
185
4-2
Prime Factorization
Main Ideas • Write the prime factorizations of composite numbers.
There are two ways that 10 can be expressed as the product of whole numbers. This can be shown by using 10 squares to form rectangles. 10
• Factor monomials.
1
2
1 10 10
New Vocabulary prime number composite number prime factorization factor tree monomial factor
5
2 5 10
a. Use grid paper to draw as many different rectangular arrangements of 2, 3, 4, 5, 6, 7, 8, and 9 squares as possible. b. Which numbers of squares can be arranged in more than one way? c . Which numbers of squares can only be arranged one way? d. What do the rectangles in part c have in common? Explain.
Prime Numbers and Composite Numbers A prime number is a whole number that has exactly two factors, 1 and itself. A composite number is a whole number that has more than two factors. Zero and 1 are neither prime nor composite. Whole Numbers
⎧ Prime ⎨ Numbers ⎩ ⎧ Composite ⎨ Numbers ⎩ Neither Prime ⎧ ⎨ nor Composite ⎩
Vocabulary Link Composite Everyday Use materials that are made up of many substances Math Use numbers having many factors
EXAMPLE
Factors
Number of Factors
2 3 5 7
1, 2 1, 3 1, 5 1, 7
2 2 2 2
4 6 8 9
1, 2, 4 1, 2, 3, 6 1, 2, 4, 8 1, 3, 9
3 4 4 3
0 1
all numbers 1
infinite 1
Identify Numbers as Prime or Composite
a. Determine whether 19 is prime or composite. Find factors of 19 by listing the whole number pairs whose product is 19. 19 = 1 × 19 The number 19 has only two factors. Therefore, 19 is a prime number. 186 Chapter 4 Factors and Fractions
Extra Examples at pre-alg.com
b. Determine whether 28 is prime or composite. Find factors of 28 by listing the whole number pairs whose product is 28.
Mental Math
28 = 1 × 28 28 = 2 × 14 28 = 4 × 7 The factors of 28 are 1, 2, 4, 7, 14, and 28. Since the number has more than two factors, it is composite.
To determine whether a number is prime or composite, you can mentally use the rules for divisibility rather than listing factors. You can review divisibility rules on pages 740–741.
Determine whether each number is prime or composite. 1A. 21
1B. 37
When a composite number is expressed as the product of prime factors, it is called the prime factorization of the number. One way to find the prime factorization of a number is to use a factor tree. Write the number that you are factoring at the top.
24
24
Choose any pair of whole number factors of 24.
· 12
2
2·3
·
4
2·3·2
·
Continue to factor any number that is not prime.
2
2
8
·
4
·
2·3
·
2·2·3
3
The factor tree is complete when you have a row of prime numbers.
Both trees give the same prime factors, except in different orders. There is exactly one prime factorization of 24. The prime factorization of 24 is 2 · 2 · 2 · 3 or 23 · 3.
EXAMPLE
Write Prime Factorization
Write the prime factorization of 36. 36
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
6
·
6
36 = 6 · 6
2 · 3 · 2 · 3 6=2·3 The factorization is complete because 2 and 3 are prime numbers. The prime factorization of 36 is 2 · 2 · 3 · 3 or 22 · 32.
Write the prime factorization of each number. Use exponents for repeated factors. 2A. 16 2B. 27 Personal Tutor at pre-alg.com Lesson 4-2 Prime Factorization
187
Factor Monomials A number such as 80 or an expression such as 8x is called a monomial. A monomial is a number, a variable, or a product of numbers and/or variables. 8 · 10 = 80
8 · x = 8x
8 and 10 are factors of 80.
8 and x are factors of 8x.
To factor a number means to write it as a product of its factors. A monomial can also be factored as a product of prime numbers and variables with no exponent greater than 1. Negative coefficients can be factored using -1 as a factor.
EXAMPLE
Factor Monomials
Factor each monomial. a. 8ab2 8ab2 = 2 · 2 · 2 · a · b2 =2·2·2·a·b·b
8=2·2·2 a · b2 = a · b · b
b. -30x3y -30x3y = -1 · 2 · 3 · 5 · x3 · y -30 = -1 · 2 · 3 · 5 = -1 · 2 · 3 · 5 · x · x · x · y x3 · y = x · x · x · y c. -16e2f 3 -16e2f 2 = -1 · 2 · 2 · 2 · 2 = -1 · 2 · 2 · 2 · 2 · e · e · f · f · f
3A. 10x2y
Example 1 (pp. 186–187)
-16 = -1 · 2 · 2 · 2 · 2 e2 · f 3 = e · e · f · f · f
3B. -18mn4
Determine whether each number is prime or composite. 1. 7
2. 23
3. 15
4. NUMBER THEORY One mathematical conjecture that is unproved states that there are infinitely many twin primes. Twin primes are prime numbers that differ by 2, such as 3 and 5. List all the twin primes that are less than 50. Example 2 (p. 187)
Write the prime factorization of each number. Use exponents for repeated factors. 5. 18
Example 3 (p. 188)
6. 39
7. 50
ALGEBRA Factor each monomial. 8. 4c2
188 Chapter 4 Factors and Fractions
9. 5a2b
10. -70xyz
HOMEWORK
HELP
For See Exercises Examples 11–18 1 19–26 2 27–38 3
Determine whether each number is prime or composite. 11. 21
12. 33
13. 23
14. 70
15. 17
16. 51
17. 43
18. 31
Write the prime factorization of each number. Use exponents for repeated factors. 19. 26
20. 81
21. 66
22. 63
23. 104
24. 100
25. 392
26. 110
ALGEBRA Factor each monomial. 27. 14w
28. 9t2
29. -7c2
30. -25z3
31. 20st
32. -38mnp
33. 28x2y
34. 21gh3
35. 13q2r2
36. 64n3
37. -75ab2
38. -120r2st3
TECHNOLOGY Mersenne primes are prime numbers in the form 2n - 1. In 2004, Josh Findlay used special software to discover the largest prime number so far, 224,036,583 - 1. Write the prime factorization of each number, or write prime if the number is a Mersenne prime. 39. 25 - 1
40. 26 - 1
41. 27 - 1
42. 28 - 1
43. CALENDARS February 3 is a prime day because the month and day (2/3) are represented by prime numbers. How many prime days are there in a non-leap year? EXTRA
PRACTIICE
See pages 768, 797. Self-Check Quiz at pre-alg.com
H.O.T. Problems
44. Is the value of n2 - n + 41 prime or composite if n = 3? 45. PACKAGING A beverage company is developing the packaging for a supercase of soda that contains 36 cans. List the arrangement of the cans that could be used for the package. (Hint: The cans can be stacked as well as arranged in a rectangular pattern one-can high.) 46. OPEN ENDED Write a monomial whose factors include -1, 5, and x. 47. FIND THE ERROR Cassidy and Francisca each factored 88. Who is correct? Explain your reasoning. Francisca 88
Cassidy 88 4 ·
22
4 · 2 · 11 88 = 4 · 2 · 11
2
8
11
·
4 · 11
2 · 2 · 2 · 11 88 = 2 · 2 · 2 · 11
48. CHALLENGE Find the prime factors of these numbers that are divisible by 12: 12, 60, 84, 132, and 180. Then, write a rule to determine when a number is divisible by 12. 49. NUMBER SENSE Find the least number that gives you a remainder of 1 when you divide it by 2, 3, 5, or 7. Lesson 4-2 Prime Factorization
189
50.
Writing in Math Use the information about prime numbers on page 186 to explain how models can be used to determine whether numbers are prime. Include the number of rectangles that can be drawn to represent a prime or a composite number and an explanation of how one model can show that a number is not prime.
51. Prime numbers are used to help keep messages sent over the Internet private. One step in the process involves multiplying two prime numbers to produce a key N. Which number could be N?
52. How many rectangles with different wholenumber dimensions can be drawn if each rectangle has an area of 30 square centimeters?
A 27
C 31
F 2
H 4
B 29
D 33
G3
J
Ài> Îä VÓ
5
53. Write (-5) · (-5) · (-5) · h · h · k using exponents. (Lesson 4-1) 54. FENCING Luis has 48 feet of fencing and is planning to make a rectangular pen for his dog. The length of the fence is 3 times as long as the width. If he uses all of the fencing, what are the dimensions of the pen? (Lesson 3-8) TIME ZONES The table shows a relationship between times in the Pacific Standard Time Zone (PST), where Seattle is located, and the Eastern Standard Time Zone (EST), where New York is located. (Lesson 3-7) 55. What time is it in New York if it is 6 P.M. in Seattle? 56. What time is it in Seattle if it is 11 P.M. in New York? 57. Write in words the relationship between PST and EST. ALGEBRA Solve each equation. Check your solution. (Lesson 3-4) n 58. _ = -4 8
59. 2x = -18
60. 30 = 6n
y 4
61. -7 = _
62. ALGEBRA Evaluate 9 + t if t = -1. (Lesson 2-2) Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 1-7) 63. outside temperature and amount of heating bill 64. size of a television screen and the number of channels it receives
PREREQUISITE SKILL Use the Distributive Property to write each expression as an equivalent expression. (Lesson 3-1) 65. 2(n + 4)
66. 5(x - 7)
67. -3(t + 4)
68. (a + 6)10
69. (b - 3)(-2)
70. 8(9 - y)
190 Chapter 4 Factors and Fractions
*-/
-/
£ *°°
{ *°°
Ó *°°
x *°°
Î *°°
È *°°
{ *°°
Ç *°°
4-3
Greatest Common Factor
Main Ideas • Find the greatest common factor of two or more numbers or monomials. • Use the Distributive Property to factor algebraic expressions.
New Vocabulary Venn diagram greatest common factor
A Venn diagram shows the relationships among sets of numbers or objects by using overlapping circles in a rectangle.
Prime Factors of 12
3
The Venn diagram at the right shows the prime factors of 12 and 20. The common prime factors are in both circles.
12 2 · 2 · 3
Prime Factors of 20 2 2
5
20 2 2 5
a. Which numbers are in both circles? b. Find the product of the numbers that are in both circles. c. Is the product also a factor of 12 and 20? d. Make a Venn diagram showing the prime factors of 16 and 28. Then use it to find the common factors of the numbers.
Greatest Common Factor Often, numbers have some of the same factors. The greatest number that is a factor of two or more numbers is called the greatest common factor (GCF). Example 1 shows several ways to find the GCF.
EXAMPLE Choosing a Method
Find the GCF
a. Find the GCF of 12 and 20.
To find the GCF of two or more numbers, it is easier to
Method 1 List the factors.
• list the factors if the numbers are small, or • use prime factorization if the numbers are large.
factors of 20: 1, 2, 4, 5, 10, 20
factors of 12: 1, 2, 3, 4, 6, 12
Common factors of 12 and 20: 1, 2, 4
The greatest common factor of 12 and 20 is 4. Method 2 Use prime factorization. 12 = 2 · 2 · 3 20 = 2 · 2 · 5
Common prime factors of 12 and 20: 2, 2
The GCF is the product of the common prime factors. 2·2=4 Again, the GCF of 12 and 20 is 4. (continued on the next page) Lesson 4-3 Greatest Common Factor
191
b. Find the GCF of 30 and 24. First, factor each number completely. Then circle the common factors. 30
24
2 · 15
2 · 12
2· 3 · 5
2·2 · 6 2·2·2 · 3
30 = 2 ·
3 ·5
24 = 2 · 2 · 2 · 3 Writing Prime Factors Try to line up the common prime factors so that it is easier to circle them.
The common prime factors are 2 and 3.
The GCF of 30 and 24 is 2 · 3 or 6. c. Find the GCF of 54, 36, and 45. 54 = 2 ·
3 · 3 ·3
36 = 2 · 2 · 3 · 3 45 =
The common prime factors are 3 and 3.
Prime Factors of 54 3
3 · 3 ·5
The GCF is 3 · 3 or 9. Prime Factors of 45
Prime Factors of 36 2 33
2
5
Find the GCF of each set of numbers. 1A. 6, 24 1B. 16, 60
1C. 10, 25, 30
TRACK AND FIELD There are 208 boys and 240 girls participating in a field day competition. a. What is the greatest number of teams that can be formed if all the teams have the same number of girls and the same number of boys? Find the GCF of 208 and 240. 208 = 2 · 2 · 2 · 2 · 13 240 = 2 · 2 · 2 · 2 · 3 · 5
The common prime factors are 2, 2, 2, and 2.
The GCF is 2 · 2 · 2 · 2 or 16. So, 16 teams can be formed. b. How many boys and girls will be on each team? Real-World Link In some events such as sprints and the long jump, if the wind speed is greater than 2 meters per second (or 4.5 miles per hour) then the time or mark cannot be considered for record purposes. Source: encarta.msn.com
boys: 208 ÷ 16 = 13
girls: 240 ÷ 16 = 15
So, each team will have 13 boys and 15 girls.
FOOD Marta is cutting a 16-inch and a 28-inch submarine sandwich for a party. 2A. How long is the longest possible piece if she cuts them all to be the same length? 2B. How many total pieces are there? Personal Tutor at pre-alg.com
192 Chapter 4 Factors and Fractions Andy Lyons/Getty Images
Factor Algebraic Expressions You can also find the GCF of two or more monomials by finding the product of their common prime factors.
EXAMPLE
Find the GCF of Monomials
Find the GCF of 16xy2 and 30xy. Completely factor each expression. 16xy2 = 2 · 2 · 2 · 2 · x · y · y 30xy = 2 · 3 · 5 ·
Circle the common factors.
x · y
The GCF of 16xy2 and 30xy is 2 · x · y or 2xy.
3. Find the GCF of 8ab and 18b2.
You can use a GCF to factor an algebraic expression such as 2x + 6.
EXAMPLE
Factor Expressions
Factor 2x + 6. First, find the GCF of 2x and 6. 2x = 2 · x 6 = 2 · 3 The GCF is 2. Now write each term as a product of the GCF and its remaining factors.
Look Back To review the Distributive Property, see Lesson 3-1.
2x + 6 = 2(x) + 2(3) = 2(x + 3) Distributive Property
4A. 4d + 8
Example 1 (pp. 191–192)
Example 2 (p. 192)
Example 3 (p. 193)
Example 4 (p. 193)
Factor each expression. 4B. 3x + 9
Find the GCF of each set of numbers. 1. 6, 8
2. 21, 45
3. 16, 56
4. 28, 42
5. 12, 24, 36
6. 6, 15, 24
7. PARADES In the parade, 36 members of the color guard are to march in front of 120 members of the high school marching band. Both groups are to have the same number of students in each row. Find the greatest number of students that could be in each row. Find the GCF of each set of monomials. 8. 2y, 10y2
9. 14n, 42n3
10. 36a3b, 56ab2
Factor each expression. 11. 3n + 9
Extra Examples at pre-alg.com
12. t2 + 4t
13. 15 + 20x Lesson 4-3 Greatest Common Factor
193
HOMEWORK
HELP
For See Exercises Examples 14–25 1 26–27 2 28–36 3 37–42 4
Find the GCF of each set of numbers or monomials. 14. 12, 8
15. 3, 9
16. 24, 40
17. 21, 14
18. 20, 30
19. 12, 18
20. 42, 56
21. 30, 35
22. 9, 15, 24
23. 20, 21, 25
24. 20, 28, 36
25. 66, 90, 150
26. QUILTING Suki is making a quilt from two different kinds of fabrics. One is 48 inches wide and the other is 54 inches wide. What are the dimensions of the largest squares she can cut from both fabrics so there is no wasted fabric? 27. DESIGN Lauren is covering the surface of an end table with equal-sized ceramic tiles. The table is 30 inches long and 24 inches wide. What is the largest square tile that Lauren can use and not have to cut any tiles? How many tiles will Lauren need? Find the GCF of each set of monomials. 28. 18, 45mn
29. 24t2, 32
30. 12x, 40x2
31. 4st, 10s
32. 5ab, 6b2
33. 7x2, 15xy
34. 14b, 56b2
35. 25k, 35j
36. 21x2y, 63xy2
37. 2x + 8
38. 3r + 12
39. 8 + 40a
40. 6 + 3y
41. 15f + 18
42. 14 + 21c
Factor each expression.
PATTERNS For Exercises 43 and 44, consider the pattern 7, 14, 21, 28, 35, . . . 43. Find the GCF of the terms in the pattern. Explain how you know. 44. Write the next two terms in the pattern. Find the GCF of each set of monomials. 45. 30a3b2, 24a2b
46. 32mn2, 16n, 12n3
NUMBER THEORY Two numbers are relatively prime if their only common factor is 1. Determine whether the numbers in each pair are relatively prime. Write yes or no. 47. 7 and 8
48. 13 and 11
49. 27 and 18
50. 20 and 25
Factor each expression. 51. k3 + k2 + 5k
52. 2x + 4y - 16
53. 5n - 10m + 25
Find possible dimensions of each rectangle, given the area. EXTRA
PRACTICE
55.
54.
56. Ài>
See pages 768, 797.
Ài> ÎX È Self-Check Quiz at pre-alg.com
194 Chapter 4 Factors and Fractions
XÓ
ÓX
Ài> Y Ó Y
H.O.T. Problems
57. OPEN ENDED Name two different numbers whose GCF is 12. 58. FIND THE ERROR Christine and Jack both found the GCF of 2 · 32 · 11 and 23 · 5 · 11. Who is correct? Explain your reasoning. Jack 2 · 32 · 11 2 · 2 · 2 · 5 · 11 GCF = 2 · 11 or 22
Christine 2 · 32 · 11 23 · 5 · 11 GCF = 11
59. CHALLENGE Can the GCF of a set of numbers be equal to one of the numbers? Give an example or a counterexample to support your answer. 60.
Writing in Math Use the information about GCF on page 191 to explain how a Venn diagram can be used to show the greatest common factor.
61. The Venn diagram shows the factors of 10x and 18x2. What is the greatest common factor of the two monomials? £äÝ
A x
B 2x
£nÝÓ X X
C x2
62. Terrell is cutting paper streamers to decorate for a party. He has a blue roll of paper 144 inches long, a red roll 192 inches long, and a yellow roll 360 inches long. If he wants to have all colors the same length, what is the longest length that he can cut?
D 2
F 24 in.
H 16 in.
G 18 in.
J 12 in.
ALGEBRA Factor each monomial. (Lesson 4-2) 63. 9n
64. 15x2
66. 22ab3
65. -5jk
67. ALGEBRA Evaluate 7x2 + y3 if x = -2 and y = 4. (Lesson 4-1) SALES An online bookstore charges a delivery fee for every book order placed on its Web site. The table shows the relationship between the book order amount and the total amount due. (Lesson 3-7) 68. What is the total if a $7 book order was placed? 69. What was the amount of the order placed if the total amount was $16?
Book Order (dollars)
Total (dollars)
1
4
2
5
3
6
4
7
Find each quotient. (Lesson 2-5) 70. -69 ÷ 23
71. 48 ÷ (-8)
72. -24 ÷ (-12)
73. -50 ÷ 5
PREREQUISITE SKILL Find each equivalent measure. (Pages 753–756) 74. 1 ft = ? in. 75. 1 yd = ? in. 76. 1 day = ? h 77. 1 m = ? cm Lesson 4-3 Greatest Common Factor
195
4-4
Simplifying Algebraic Fractions
Main Ideas • Simplify fractions using the GCF.
You can use a fraction to compare a part of something to a whole. The figures below show what part 15 minutes is of 1 hour.
• Simplify algebraic fractions.
11
12
1 2
10
New Vocabulary
11
simplest form algebraic fraction
8 6
2
15 60
2 3
8
4 7
5
6
1
9
4 7
12
10
3
8
5
15 of 60 parts are shaded.
11
9
4 7
1
10
3
9
12
3 of 12 parts are shaded.
3 12
6
5
1 of 4 parts is shaded.
1 4
a. Are the three fractions equivalent? Explain your reasoning. b. Which figure is divided into the least number of parts? c. Which fraction would you say is written in simplest form? Why?
Simplify Numerical Fractions A fraction is in simplest form when the GCF of the numerator and the denominator is 1. Fractions in Fractions not in Simplest Form Simplest Form 17 _1 , _1 , _3 , _
3 _ 6 _ _ , 15 , _ , 5
4 3 4 50
12 60 8 20
One way to write a fraction in simplest form is to write the prime factorizations of the numerator and the denominator. Then divide the numerator and denominator by the GCF.
EXAMPLE
Simplify Fractions
Prime Factors of 9
9 Write _ in simplest form. 12
9=3·
3
12 = 2 · 2 · 3
Factor the numerator. Factor the denominator.
Use a Venn Diagram
The GCF of 9 and 12 is 3.
To simplify fractions, let one circle in a Venn diagram represent the factors of the numerator and the other circle represent the factors of the denominator. The product of factors in the intersection is the GCF.
12
9÷3 9 _ =_ 12 ÷ 3 3 =_ 4
3 3 2 2
Divide the numerator and the denominator by the GCF. Simplest form
Prime Factors of 12
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 16 8 1A. _ 1B. _
196 Chapter 4 Factors and Fractions
20
9
The division in Example 1 can be represented in another way. 1
9 3·3 _ =_
The slashes mean that the numerator and the denominator are both divided by the GCF, 3.
2·2·3
12
1
3 3 =_ or _ Simplify. 2·2
EXAMPLE
4
Simplify Fractions
Write each fraction in simplest form. 12 a. _ 48
1
1
1
2·2·3 12 _ = __
Divide the numerator and denominator 1 by the GCF, 2 · 2 · 3.
2·2·2·2·3
48
Interactive Lab pre-alg.com
1
1 =_
1
Simplify.
4
17 b. _
30 17 _ is in simplest form because the GCF of 17 and 30 is 1. 30
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 16 2A. _
24 2B. _
45
40
Eighty-eight feet is what part of 1 mile? 1 C _
1 A _
15
60 _ B 1 30
1 D _ 5
Read the Test Item Estimation You can solve some problems without much calculating by estimating your answer. 100 88 1 _ = _ or _ 5280 5000 50
The phrase what part indicates a relationship that can be written as a fraction. You need to write a fraction comparing 88 feet to the number of feet in 1 mile. Solve the Test Item 88 There are 5280 feet in 1 mile. Write the fraction _ in simplest form. 5280
1
1
1
1
88 2 · 2 · 2 · 11 _ = __ 2 · 2 · 2 · 2 · 2 · 3 · 5 · 11
5280
1
1
1
Divide the numerator and denominator by the GCF, 2 · 2 ?· 2 · 11.
1
1 =_ 60
1 So, 88 feet is _ of a mile. The answer is A. 60
Extra Examples at pre-alg.com
(continued on the next page) Lesson 4-4 Simplifying Algebraic Fractions
197
CHECK You can check whether your answer is correct by solving the problem in a different way. Divide the numerator and denominator by common factors until the fraction is in simplest form. 88 44 _ =_ 5280
2640 22 =_ 1320 11 1 =_ or _ 660 60
3. Six hundred sixteen yards is what part of 1 mile? (Hint: 1 mi = 1760 yd) 6 F_
1 G_
25
3 H_
4
7 J_ 20
10
Personal Tutor at pre-alg.com
Simplify Algebraic Fractions A fraction with variables in the numerator or denominator is called an algebraic fraction. Algebraic fractions can also be written in simplest form.
EXAMPLE Check Reasonableness of Results You can check whether your answers are reasonable. In Example 4a, you can see that the variable y does not appear in the final answer since y can be divided into both the numerator and denominator. 21x2y
_
Simplify Algebraic Fractions
Simplify each fraction. If the fraction is already in simplest form, write simplified. 2
21x y a. _
3
abc b. _ 2
35xy
ab
1
2
1
1
21x y 3·7·x·x·y _ = __ 5·7·x·y
35xy
1
1
(pp. 196–197)
(p.198)
Multiply.
3
xyz
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 9 2. _
25 4. _
5 3. _
15
40
11
64 5. _ 68
6. MULTIPLE CHOICE Nine inches is what part of 1 yard? 1 A_
1 B _
8
Example 4
Factor.
1
xy 4B. _2
28ab 4A. _ 2
14
(pp.197–198)
a·a·b
1
c =_ a
Simplify.
5
2 1. _
Example 3
a2b
1
42ab
Examples 1, 2
abc3 a·b·c·c·c _ = __ 3
3x =_
35xy
1 1
Factor out the GCF, 7 · x ?· y.
1 C_
5
1 D_ 2
4
ALGEBRA Simplify each fraction.If the fraction is already in simplest form, write simplified. x 7. _ 3 x
198 Chapter 4 Factors and Fractions
8a2 8. _ 16a
12c 9. _ 15d
24 10. _ 5k
25mn 11. _ 65n
HOMEWORK
HELP
For See Exercises Examples 12–21 1, 2 22–29 4 42–44 3
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 3 12. _
18 18 17. _ 44
10 13. _
15 14. _
12 16 18. _ 64
8 15. _
17 16. _
36 34 20. _ 38
21 30 19. _ 37
20 17 21. _ 51
ALGEBRA Simplify each fraction. If the fraction is already in simplest form, write simplified. a 22. _4
a 4k _ 26. 19m
3
y 23. _ y
12m 24. _
15m _ 28. 16n2 18n p
4t 27. _2 64t
40d 25. _ 42d
28z3 29. _ 16z
30. MEASUREMENT Twelve ounces is what part of a pound? (Hint: 1 lb = 16 oz) 31. ANALYZE TABLES The table shows the number of Nebraska tornadoes that occurred in May and the total for selected years. What fraction of tornadoes occurred in May for each year? Write each fraction in simplest form.
9EAR
-AY 4OTAL
9EAR 4OTAL
3OURCE .EBRASKA 3EVERE 7EATHER
Real-World Career Musician A musician uses math to increase or decrease the tension in the strings of his or her instrument. The pitch is the frequency at which an instrument’s string vibrates when it is struck.
32. AIRCRAFT A model of Lindbergh’s Spirit of St. Louis has a wingspan of 18 inches. The wingspan of the actual airplane is 46 feet. Write a fraction in simplest form comparing the wingspan of the model and the wingspan of the actual airplane. (Hint: Convert 46 feet to inches.) MUSIC For Exercises 33–35, use the following information. Musical notes C and A sound harmonious together because of their frequencies, or vibrations. The fraction that is formed by the two frequencies can be simplified, as shown below. C 264 3 _ =_ or _
For more information, go to pre-alg.com.
A
440
5
When a fraction formed by two frequencies cannot be simplified, the notes sound like noise. Determine whether each pair of notes would sound harmonious together. Explain why or why not. 33. E and A EXTRA
PRACTICE
34. D and F
Note
Frequency (hz)
C D E F G A B C
264 294 330 349 392 440 494 528
35. first C and last C
36. TIME Fifteen hours is what part of one day?
See pages 769, 797. Self-Check Quiz at pre-alg.com
37. FIND THE DATA Refer to the United States Data File on pages 18–21 of your book. Choose some data and write a real-world problem in which you would simplify fractions. Lesson 4-4 Simplifying Algebraic Fractions
Geoff Butler
199
H.O.T. Problems
38. OPEN ENDED Write a numerical fraction and an algebraic fraction in simplest form and a numerical fraction and an algebraic fraction not in simplest form. 39. Which One Doesn’t Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. 6y _
_4
5
_1
10 _ 12
x2
7
1
23 23 2 40. CHALLENGE Is it true that _ =_ or _ ? Explain why or why not. 53
Writing in Math
41.
53 1
5
Explain how simplified fractions are useful in representing measurements. Include an explanation of how measurements represent parts of a whole and examples of fractions that represent measurements.
42. Which store offers the best buy? A A B B
Store
Potatoes
A
18 for $12
B
30 for $24
C C
C
36 for $30
DD
D
42 for $30
43. In a pen factory, an average of 5 pens out of every p pens tested are rejected. What fraction of those p pens is NOT rejected? 5 F _ p
p G _ 5
p-5 H _ 5
44. Ninety-six centimeters is what part of a meter? 3 A _ 5
4 B _ 5
23 C _ 25
Find the GCF of each set of numbers or monomials. (Lesson 4-3) 45. 9, 15
46. 4, 12, 10
47. 40x2, 16x
48. 25a, 30b
Determine whether each number is prime or composite. (Lesson 4-2) 49. 13
50. 34
51. 99
52. 79
ALGEBRA Write and solve an equation to find each number. (Lesson 3-3) 53. The sum of a number and 9 is -2.
54. The sum of -5 a number and is -15.
55. GEOMETRY The area of a trapezoid is the product of one half the height and the sum of both bases. If h is the height, b1 is one base, and b2 is the second base, write an expression for the area of the trapezoid. (Lesson 1-3)
PREREQUISITE SKILL For each expression, use parentheses to group the numbers together and to group the powers with like bases together. (Lesson 1-4) Example: a · 4 · a3 · 2 = (4 · 2)(a · a3) 56. 6 · 7 · k3
57. s · t2 · s · t
58. b · 5 · 10 · b4
59. 3 · x4 · (-5) · x2
60. 5 · n3 · p · 2 · n · p
61. 12 · 15 · a · 9 · a5 · c3
200 Chapter 4 Factors and Fractions
p-5 J _ p
24 D_ 25
Powers The phrase the quantity is used to indicate parentheses when reading expressions. Recall that an exponent indicates the number of times that the base is used as a factor. Suppose you are to write each of the following in symbols. Words
Symbols
Examples (Let x = 2.)
3x2 =
3 · 22 = 3 · 4 Evaluate 22. = 12 Multiply 3 · 4.
3x2
three times x squared
three times x the quantity squared
(3x)2 = (3 · 2)2 = 62 Evaluate 3 · 2. = 36 Square 6.
(3x)2
In the expression (3x)2, parentheses are used to show that 3x is used as a factor twice. (3x)2 = (3x)(3x) The quantity can also be used to describe division of monomials. Words
Symbols
Examples (Let x = 2.)
8 8 _ =_ x2
8 _
eight divided by x squared
x2
22 8 = _ Evaluate 22. 4 = 2 Divide 8 ÷ 4.
8x = 82 2
eight divided by x the quantity squared
8x
2
2
= 42 Evaluate 8 ÷ 2. = 16 Square 4.
Exercises State how you would read each expression. 5 _
1. 4a2
2. (10x)5
3.
6. (a - b)4
7. a - b4
a 8. _4
n3 b
4 2 4. _r
5. (m + n)3
c
8 10. _2
9. (4c2)3
3
Determine whether each pair of expressions is equivalent. Write yes or no. 11. 4ab5 and 4(ab)5
12. (2x)3 and 8x3
13. (mn)4 and m4 · n4
14. c3d3 and cd3
2 x 15. _2 and x y
2 n2 16. _ and n 2 r
y
r
Reading Math Powers
201
CH
APTER
4
Mid-Chapter Quiz Lessons 4-1 through 4-4
1. ALGEBRA Evaluate b2 - 4ac if a = -1, b = 5, and c = 3. (Lesson 4-1)
Factor each monomial. (Lesson 4-2)
2. MULTIPLE CHOICE The number of acres consumed by a forest fire triples every two hours. Which expression represents the number of acres consumed after 1 day? (Lesson 4-1)
11. -23n3
Hours
2
4
6
8
10
Acres Consumed
31
32
33
34
35
A 310 acres
C 318 acres
B 312 acres
D 324 acres
3. Write his reward on each of the first three days as a power of 2. 4. Write his reward on the 8th day as a power of 2. Then evaluate. Write the prime factorization of each number. Use exponents for repeated factors. (Lesson 4-2) 6. 99
10. 18st 12. 30cd2
13. BAKE SALE Joanna baked 81 cookies and 54 cupcakes for the bake sale. She wants to place the same number of cookies and the same number of cupcakes in a plastic bag. What is the maximum number of bags that she can make if she uses all of the cookies and cupcakes? (Lesson 4-3) Find the GCF of each set of numbers or monomials. (Lesson 4-3)
LITERATURE In a story, a knight received a reward for slaying a dragon. He received 1 cent on the first day, 2 cents on the second day, 4 cents on the third day, and so on, continuing to double the amount for 30 days. (Lesson 4-1)
5. 42
9. 77x
7. 64
8. MULTIPLE CHOICE A kitchen floor with the dimensions shown is to be tiled. If the tiles are only available in dimensions that are prime numbers, which set of tile dimensions could NOT be used to tile the floor? (Lesson 4-2)
14. 18, 45
15. 22, 21
16. 16, 40
17. 10, 12m
18. 3x, 18x2
19. 15g, 35h
Factor each expression. (Lesson 4-3) 20. 9s + 18
21. 6y + 21
22. 60 + 15h
23. 9z - 99
24. 21x - 63
25. 18 - 12a
26. MULTIPLE CHOICE Three hundred thirty yards is what part of 1 mile? (Lesson 4-4) 1 A 16
2 27. x3
x
G 2 ft by 3 ft
J 3 ft by 3 ft
202 Chapter 4 Factors and Fractions
20d
30. ANIMALS The table shows the average amount of food each animal can eat in a day and its average weight. What fraction of its weight can each animal eat per day? (Lesson 4-4)
elephant
H 2 ft by 5 ft
5c3d 29. 2
2 28. 27n 15
Animal
F 2 ft by 2 ft
D 1 4
ALGEBRA Simplify each fraction. If the fraction is already in simplest form, write simplified. (Lesson 4-4)
£Ó vÌ
Óä vÌ
3 C 16
B 1 8
hummingbird polar bear tiger
Daily Weight of Amount Animal of Food 450 lb 9000 lb 2g
3g
25 lb 20 lb
1500 lb 500 lb
Source: Animals as Our Companions, Wildlife Fact File
4-5
Multiplying and Dividing Monomials
Main Ideas • Multiply monomials. • Divide monomials.
For each increase of 1 on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater. So, an earthquake of magnitude 4 has seismic waves that are 10 times greater than that of a magnitude 3 earthquake.
Richter Scale
Times Greater than Magnitude 3 Earthquake
Written Using Powers
4
10
101
5
10 ⴛ 10 ⴝ 100
101 ⴛ 101 ⴝ 102
6
10 ⴛ 100 ⴝ 1000
101 ⴛ 102 ⴝ 103
7
10 ⴛ 1000 ⴝ 10,000
101 ⴛ 103 ⴝ 104
8
10 ⴛ 10,000 ⴝ 100,000
101 ⴛ 104 ⴝ 105
a. Examine the exponents of the factors and the exponents of the products in the last column. What do you observe? b. MAKE A CONJECTURE Write a rule for determining the exponent of the product when you multiply powers with the same base. Test your rule by multiplying 23 · 24 using a calculator.
Multiply Monomials Recall that exponents are used to show repeated multiplication. You can use the definition of exponent to help find a rule for multiplying powers with the same base. 3 factors
4 factors
23 · 24 = (2 · 2 · 2) · (2 · 2 · 2 · 2) = 27
7 factors
Notice the sum of the original exponents and the exponent in the final product. This relationship is stated in the following rule. Common Misconception
Product of Powers Multiply powers with the same base by adding their exponents.
When multiplying powers, do not multiply the bases.
Words
32 · 34 = 36, not 96
Example 32 · 34 = 32 + 4 or 36
Symbols a m · an = a m + n
Lesson 4-5 Multiplying and Dividing Monomials
203
EXAMPLE
Multiply Powers
Find 73 · 7. 73 · 7 = 73 · 71 7 = 71 = 73 + 1
The common base is 7.
= 74
Add the exponents.
Find each product. 1A. 52 · 53
1B. 24 · 26
Monomials can also be multiplied using the rule for the product of powers.
EXAMPLE
Multiply Monomials
Find each product. a. x5 · x2 x5 · x2 = x 5 + 2 = x7
The common base is x. Add the exponents.
b. (-4n3)(2n6) Look Back
(-4n3)(2n6) = (-4 · 2)(n3 · n6)
To review the Commutative and Associative Properties of Multiplication, see Lesson 1-4.
Group the coefficients and variables.
= (-8)(n3 + 6)
The common base is n.
= -8n9
Add the exponents.
2A. y6 · y3
2B. (5a2)(-3a4)
Divide Monomials You can also find a rule for quotients of powers. 26 = 2 · 2 · 2 · 2 · 2 · 2 1 2 2
6 factors 1 factor
1
2·2·2·2·2·2 = 2
1
= 25
5 factors
Divide the numerator and the denominator by the GCF, 2. Simplify.
Compare the difference between the original exponents and the exponent in the final quotient. This relationship is stated in the following rule. Quotient of Powers
BrainPOP® pre-alg.com
Words
Divide powers with the same base by subtracting their exponents.
Symbols
am _ = am - n, where a ≠ 0
an 45 Example _ = 45 - 2 or 43 42
204 Chapter 4 Factors and Fractions
EXAMPLE
Divide Powers
Find each quotient. 5
y b. _3
57 a. _ 4
5 57 _ = 57 - 4 54
=
53
The common base is 5.
= y5 - 3
The common base is y.
y2
Subtract the exponents.
=
b7 3B. _ 6
3
How Many/ How Much How many times faster indicates that division is to be used to solve the problem. If the question had said how much faster, then subtraction (1010 - 109) would have been used to solve the problem.
y3
Subtract the exponents.
39 3A. _ 2
Reading Math
y y5 _
b
COMPUTERS The table compares the processing speeds of a specific type of computer in 1999 and in 2004. Find how many times faster the computer was in 2004 than in 1999.
Year
(instructions per second)
Write a division expression to compare the speeds. 1010 _ = 1010 - 9 109
= 101 or 10
Processing Speed
1999
109
2004
1010
Subtract the exponents. Simplify.
So, the computer was 10 times faster in 2004 than in 1999.
Source: The Intel Microprocessor Quick Reference Guide
4. TRAVEL The table compares the number of people who drove to work versus the number of people who walked to work in Wyoming in 2004. How many times more people drove than walked to work?
Mode of Transportation
Number of People
Drove
105
Walked
103
Source: U.S. Bureau of the Census
Personal Tutor at pre-alg.com
Examples 1–3 (pp. 204–205)
Find each product or quotient. Express using exponents. 1. 93 · 92
2. 114 · 116
3. 6 · 66
4. a · a5
5. (n4)(n4)
6. -3x2(4x3)
38
7. _5 3
Example 4 (p. 205)
105
8. _3 10
a10 9. _ 6 a
10. EARTHQUAKES In 2005, an earthquake measuring 5 on the Richter scale struck the Philippines. Four days later, an earthquake of magnitude 3 struck Southern Alaska. How many times greater were the seismic waves in the Philippines than in Alaska? (Hint: Let 105 and 103 represent the strength of the earthquakes, respectively.)
Extra Examples at pre-alg.com
Lesson 4-5 Multiplying and Dividing Monomials
205
HOMEWORK
HELP
For See Exercises Examples 11–14 1 15–22, 31 2 23–30, 32 3 33–37 4
Find each product or quotient. Express using exponents. 11. 33 · 32
12. 6 · 67
13. 94 · 95
14. 104 · 103
15. d4 · d6
16. n8 · n
17. t2 · t4
18. a6 · a6
19. 2y · 9y4
20. (5r 3)(4r4)
21. (10x)(4x7)
22. 6p7 · 9p7
25. (-2)6 ÷ (-2)5
26. 1010 ÷ 102
55
23. _2 5
27. b6 ÷ b3
84
24. _3 8 a8 28. _ a8
5
(-x) 29. _
30. m20 ÷ m8
(-x)
31. the product of nine to the fourth power and nine cubed 32. the quotient of k to the fifth power and k squared PHYSICAL SCIENCE For Exercises 33–35, use the information at the left. The pH of a solution describes its acidity. Each one-unit decrease in the pH means that the solution is 10 times more acidic. For example, a pH of 4 is 10 times more acidic than a pH of 5. 33. How much more acidic is vinegar than baking soda? 34. Suppose the pH of a lake is 5 due to acid rain. How much more acidic is the lake than water? 35. Cola is 104 times more acidic than water. What is the pH value of cola?
Real-World Link The pH values of different kitchen items are shown below. Item pH lemon Juice 2 vinegar 3 tomatoes 4 water 7 baking soda 9 Source: Biology, Raven
EXTRA
PRACTIICE
See pages 769, 797. Self-Check Quiz at pre-alg.com
H.O.T. Problems
GEOMETRY For Exercises 36 and 37, use the information in the figures. 36. How many times greater is the length of the edge of the larger cube than the smaller one? 37. How many times greater is the volume of the larger cube than the smaller one?
Volume ⴝ 23 cubic units
Find each missing exponent. • 39. t2 = t14 38. (4•)(43) = 411
Volume ⴝ 26 cubic units
135 40. =1 13•
t
41. What is the product of 73, 75, and 7? 42. Find a4 · a6 ÷ a2. ARTS AND CRAFTS For Exercises 43 and 44, use the information below. When a piece of paper is cut in half, the result is two smaller pieces of paper. When the two smaller pieces are stacked and then cut, four pieces of paper are made. The number of resulting sheets of paper after c cuts is 2c. 43. How many more pieces of paper are there if a piece of paper is cut and stacked 8 times than when a piece of paper is cut and stacked 5 times? 44. Notebook paper usually stacks about 500 sheets to the inch. How thick would your stack be if you were able to make 10 cuts? Find each product or quotient. Express using exponents. 45. ab5 · 8a2b5
46. 10x3y · (-2xy2)
3
5
n (n ) 47. _ 2 n
s7 48. _ 2 s·s
49. OPEN ENDED Write a multiplication expression whose product is 53. 50. CHALLENGE Use the laws of exponents to show why the value of any nonzero number raised to the zero power equals 1.
206 Chapter 4 Factors and Fractions Laura Sifferlin
51. REASONING Determine whether the statement is true or false. If true, explain your reasoning. If false, give a counterexample. For any integer a, (-a)2 = -a2. 52.
Writing in Math Use the data about earthquakes on page 203 to explain how powers of monomials are useful in comparing earthquake magnitudes. Include a comparison of two earthquakes of different magnitudes by using the Quotient of Powers rule.
53. How many times as intense is a rock band as a noisy office?
Sound intensity is measured in decibels. The decibel scale is based on powers of ten as shown in the table. Sound
Decibels
Intensity
rock band noisy office normal conversation whispering
120 60
1012 106
50
105
20
102
A 102
C 1012
B 106
D 1072
54. How many times as intense is a noisy office as a person whispering? F 10,000
H 100
G 1000
J 10
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 12 55. _ 40
20 56. _ 53
8n2 57. _
6x3 58. _ 4
32n
4x y
Find the greatest common factor of each set of numbers or monomials. (Lesson 4-3) 59. 36, 4
60. 18, 28
62. 9a, 10a3
61. 42, 54
63. ALGEBRA A number is divided by -6, and the result is 24. What is the original number? (Lesson 3-4) 64. Evaluate |a| - |b| · |c| if a = -16, b = 2, and c = 3. (Lesson 2-1)
PREREQUISITE SKILL Evaluate each expression if x = 10, y = -5, and z = 4. Write as a fraction simplest form. (Lesson 1-3) 1 66. _ x _ 68. 1 zy
z 67. _ 100 1 69. _
Energy Used 160 Electricity Used (megawatts)
65. ENERGY The graph shows the high temperature and the amount of electricity used during each of fifteen summer days. Do the data show a positive, negative, or no relationship? Explain. (Lesson 1-7)
150 140 130 120 110 100 0
85
90
95
100
High Temperature (˚F)
(z)(z)(z)
Lesson 4-5 Multiplying and Dividing Monomials
207
EXTEND
4-5
Algebra Lab
A Half-Life Simulation
A radioactive material such as uranium decomposes or decays in a regular manner best described as a half-life. A half-life is the time it takes for half of the atoms in the sample to decay.
ACTIVITY
COLLECT THE DATA
Number of Half-Lives
Step 1 Place 50 pennies heads up in a shoebox. Put the lid on the box and shake it up and down one time. This simulates one half-life.
Number of Pennies That Remain
1 2 3
Step 2 Open the lid of the box and remove all the pennies that are now tails up. In a table like the one at the right, record the number of pennies that remain.
4 5
Step 3 Put the lid back on the box and again shake the box up and down one time. This represents another half-life. Step 4 Open the lid. Remove all the tails up pennies. Count the pennies that remain. Step 5 Repeat the half-life of decay simulation until fewer than five pennies remain in the shoebox.
ANALYZE THE RESULTS 1. On grid paper, draw a coordinate grid in which the x-axis represents the number of half-lives and the y-axis represents the number of pennies that remain. Plot the points (number of half-lives, number of remaining pennies) from your table. 2. Describe the graph of the data. After each half-life, you expect to remove about one-half of the pennies. So, you expect about one-half to remain. The expressions at the right represent the average number of pennies that remain if you start with 50, after one, two, and three half-lives.
one half-life: two half-lives: three half-lives:
50 1 1 = 50 1 2 2 2 1 1 1 50 = 50 1 2 2 2 2 1 50 1 = 50 1 2 2
3. MAKE A CONJECTURE Use the expressions to predict how many pennies remain after three half-lives. Compare this number to the number in the table above. Explain any differences. 4. MAKE A CONJECTURE Suppose you started with 1000 pennies. Predict how many pennies would remain after three half-lives. 208 Chapter 4 Factors and Fractions Latent Image
2
3
4-6
Negative Exponents
Main Ideas • Write expressions using negative exponents.
Power
Value
26
64
25
32
24
16
23
8
22
4
21
2
?
?
?
?
Copy the table at the right. a. Describe the pattern of the powers in the first column. Continue the pattern by writing the next two powers in the table.
• Evaluate numerical expressions containing negative exponents.
b. Describe the pattern of values in the second column. Then complete the second column. c. Verify that the powers you wrote in part a are equal to the values that you found in part b. d. Determine how 3-1 should be defined.
Negative Exponents Extending the pattern at the right 1 . shows that 2-1 can be defined as _
22 = 4
You can apply the Quotient of Powers rule and the x3 and write a general rule definition of a power to _ x5 about negative powers.
20 = 1
÷2
2-1 = 1 2
÷2
21 = 2
2
Method 1 Quotient of Powers
Method 2 Definition of Power 1
1
1
x3 x·x·x _ = __
x3 _ = x3 - 5
x· x · x · x · x
x5
x5
=
÷2
1 1
1
1 1 =_ or _
x-2
x·x
x3
x2
1 . Since _5 cannot have two different values, you can conclude that x-2 = _ x x2 This suggests the following definition.
Negative Exponents Symbols Example
a-n = _ n , for a ≠ 0 and any whole number n.
1 a 1 5-4 = _ 54
EXAMPLE
Use Positive Exponents
Write each expression using a positive exponent. a. 6-2
b. x -5
1 6-2 = _ Definition of negative 2 6
1A. 3-5 Extra Examples at pre-alg.com
1 x-5 = _ Definition of negative 5 x
exponent
exponent
1B. y-3 Lesson 4-6 Negative Exponents
209
EXAMPLE
Use Negative Exponents
1 Write _ as an expression using a negative exponent. 9
1 _1 = _ 9
Find the prime factorization of 9.
3·3 _ = 12 3
Definition of exponent
= 3-2
Definition of negative exponent
1 2. Write _ as an expression using a negative exponent. 25
Negative exponents are often used in science when dealing with very small numbers. Usually the number is a power of ten.
ANIMALS Geckos have tiny hairs on the bottom of their feet called setae. These setae are about 0.000001 meter long. Write the decimal as a fraction and as a power of ten. The digit 1 is in the millionths place. 1 0.000001 = _ Real-World Link Most geckos lack movable eyelids and the largest species of geckos measure 14 inches.
Write the decimal as a fraction.
1,000,000 1 =_ 106
1,000,000 = 106
= 10-6
Definition of negative exponent
1 Therefore, 0.000001 is _ as a fraction and 10-6 as a power of 10. 1,000,000
Source: encyclopedia.com
3. CHEMISTRY A hydrogen atom is only 0.00000001 centimeter in diameter. Write the decimal as a fraction and as a power of 10. Personal Tutor at pre-alg.com
Evaluate Expressions Algebraic expressions containing negative exponents can be written using positive exponents and then evaluated.
EXAMPLE
Algebraic Expressions with Negative Exponents
Evaluate n-3 if n = 2. n-3 = 2-3 1 =_ 23 1 =_ 8
Replace n with 2. Definition of negative exponent Find 23.
4. Evaluate x-4 if x = 3. 210 Chapter 4 Factors and Fractions Kim Taylor/DK Limited/CORBIS
Example 1 (p. 209)
Example 2 (p. 210)
Write each expression using a positive exponent. 1. 5-2
2. (-7)-1
1 6. _ 2
Example 4
HOMEWORK
1 7. _
1 8. _ 8
49
9
3
(p. 210)
4. n-2
Write each fraction as an expression using a negative exponent other than -1. 1 5. _ 4
Example 3
3. t-6
9. MEASUREMENT A unit of measure called a micron equals 0.001 millimeter. Write this number using a negative exponent. ALGEBRA Evaluate each expression if a = 2 and b = -3.
(p. 210)
10. a-5
HELP
Write each expression using a positive exponent.
For See Exercises Examples 14–25 1 26–33 2 34–36 3 37–40 4
11. b-3
12. (ab)-2
13. 2b
14. 4-1
15. 5-3
16. (-6)-2
17. (-3)-3
18. 3-5
19. 10-4
20. p-1
21. a-10
22. d-3
23. q-4
24. b-15
25. r-20
Write each fraction as an expression using a negative exponent other than -1. 1 26. _ 4
1 27. _ 5
1 28. _ 3
5 1 31. _ 81
9
1 30. _ 100
1 29. _ 2
8 1 32. _ 27
13 1 33. _ 16
34. BIRDS A mockingbird uses about 5-4 Joules of energy to sing a song. Write the amount of energy the bird uses as an expression using a positive exponent and as a decimal. PHYSICAL SCIENCE A nanometer is equal to a billionth of a meter. The visible range of light waves ranges from 400 nanometers (violet) to 740 nanometers (red).
400 nm
Real-World Link The wavelengths of X rays are between 1 and 10 nanometers. Source: Biology, Raven
430 nm
500 nm
560 nm
600 nm
650 nm
740 nm
35. Write one billionth of a meter as a fraction and as an expression with a negative exponent. 36. Use the information at the left to express the greatest wavelength of an X ray in meters. Write the expression using a negative exponent. ALGEBRA Evaluate each expression if w = -2, x = 3, and y = -1. 37. x-4
38. w-7
39. 8w
40. (xy)-6
Lesson 4-6 Negative Exponents Getty Images
211
41. ANALYZE TABLES Consider the pattern in the table in which the exponents are integers. If the pattern continues, what is the value of the eighth term in the pattern? Expression
23
22
21
20
Value
8
4
2
1
Write each decimal using a negative exponent. 42. 0.1
43. 0.01
44. 0.0001
45. 0.00001
1 -inch long can jump about 8 inches high. Write 46. ANIMALS A common flea _ 16 each number as an exponential expression with a base of 2. Then find how many times its body size a flea can jump.
47. MEDICINE Which type of molecule in the table has a greater mass? How many times greater is it than the other type?
EXTRA
PRACTIICE
iVÕi
>ÃÃ }®
PENICILLIN
INSULIN
See pages 769, 797. Self-Check Quiz at pre-alg.com
Use the Product of Power and Quotient of Power rules to simplify each expression. 48. x-2 · x-3 6
y 51. _ -10 y
H.O.T. Problems
4 50. x7
49. r-5 · r9
x 36s3t5 53. 12s6t-3
a4b-4 52. _ -2 ab
54. OPEN ENDED Write a convincing argument that 30 = 1 using the fact that 34 = 81, 33 = 27, 32 = 9, and 31 = 3. 1 . Does it increase or decrease as the 55. REASONING Investigate the fraction _ 2n value of n increases? Explain.
56. CHALLENGE Using what you learned about exponents, is (x3)-2 = (x-2)3? Why or why not? NUMBER SENSE Numbers can also be expressed in expanded form. Example 1: 13,548 = 10,000 + 3000 + 500 + 40 + 8 = (1 × 104) + (3 × 103) + (5 × 102) + (4 × 101) + (8 × 100) Example 2: 0.568 = 0.5 + 0.06 + 0.008 = (5 × 10-1) + (6 × 10-2) + (8 × 10-3) Write each number in expanded form. 57. 5931 61.
58. 29,607
59. 0.173
60. 0.5875
Writing in Math Use the information about negative exponents on page 209 to explain how they represent repeated division. Illustrate your reasoning with an example of a power containing a negative exponent written in fraction form.
212 Chapter 4 Factors and Fractions
62. How many square centimeters does one square millimeter equal? (Hint: 1 cm = 10 mm) A 10-1 B 10-2
£ V
C 10-3 D
63. A nurse draws a sample of blood. A cubic millimeter of the blood contains 223 white blood cells and 225 red blood cells. Compare the number of white blood cells to the number of red blood cells as a fraction.
£ V
103
10,648 F _
1 H_
484 G_
1 J _
1
484
1
10,648
ALGEBRA Find each product or quotient. Express using exponents. (Lesson 4-5) 64. 36 · 3
65. x2 · x4 3
9
y 67. _2
55 66. _ 2
16n 68. ALGEBRA Write _ in simplest form. (Lesson 4-4)
y
5
8n
69. CARPENTRY Danielle is helping her father make shelves to store her sports equipment in the garage. How many shelves measuring 12 inches by 16 inches can be cut from a 48-inch by 72-inch piece of plywood so that there is no waste? (Lesson 4-3) 70. KEYBOARDING Keyboarding speed can be determined by using the formula w - 10r where s represents the speed of words typed per minute, w s=_ m represents the number of words typed, r represents the number of errors, and m represents the total number of minutes typed. If Esteban received a keyboard speed of 80 words per minute and typed 530 words in 6 minutes, how many errors did he make? (Lesson 3-8) ALGEBRA Use the Distributive Property to rewrite each expression. (Lesson 3-1) 71. 8(y + 6)
72. -5(a - 10)
73. (9 + k)(-2)
SCHOOL For Exercises 75 and 76, use the table that shows the heights and grade point averages of the students in Mrs. Stanley’s class. (Lesson 1-7) 75. Make a scatter plot of the data. 76. Does there appear to be a relationship between the scores and the heights? Explain.
74. (n - 3)5 Name
Height (in.)
GPA
Regina
66
3.6
Michael
61
3.2
Latisha
59
3.9
Simon
64
2.8
Maurice
61
3.8
Timothy
65
3.1
Ivan
70
2.6
Helen
64
2.2
Eduardo
65
4.0
PREREQUISITE SKILL Find each product. (Pages 747–748) 77. 7.2 × 100
78. 1.6 × 1000
79. 4.05 × 10
80. 0.05 × 1000
81. 3.8 × 0.01
82. 5.0 × 0.0001
83. 9.24 × 0.1
84. 11.64 × 0.001
Lesson 4-6 Negative Exponents
213
4-7
Scientific Notation
Main Ideas • Express numbers in standard form and in scientific notation. • Compare and order numbers written in scientific notation.
New Vocabulary standard form scientific notation
A compact disc or CD has a single spiral track that stores data. It circles from the inside of the disc to the outside. If the track were stretched out in a straight line, it would be 0.5 micron wide and over 5000 meters long. a. Write the track length in millimeters.
Track Length
Track Width
5000 meters
0.5 micron
b. Write the track width in millimeters. (1 micron = 0.001 millimeter)
Scientific Notation Numbers like 5,000,000 and 0.0005 are in standard form because they do not contain exponents. However, when you deal with very large numbers like 5,000,000 or very small numbers like 0.0005, it is difficult to keep track of the place value. Numbers such as these can be written in scientific notation. Scientific Notation Words
A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.
Symbols
a × 10n, where 1 ≤ a < 10 and n is an integer
Examples 5,000,000 = 5.0 × 106
Powers of Ten To multiply by a power of 10, • move the decimal point to the right if the exponent is positive, and • move the decimal point to the left if the exponent is negative. In each case, the exponent tells you how many places to move the decimal point.
EXAMPLE
Express Numbers in Standard Form
Express each number in standard form. a. 3.78 × 106 3.78 × 106 = 3.78 × 1,000,000 106 = 1,000,000 = 3,780,000 Move the decimal point 6 places to the right. b. 5.1 × 10-5 5.1 × 10-5 = 5.1 × 0.00001 10-5 = 0.00001 = 0.000051 Move the decimal point 5 places to the left. 1A. 5.94 × 107
214 Chapter 4 Factors and Fractions Getty Images
0.0005 = 5.0 × 10-4
1B. 1.3 × 10-3 Extra Examples at pre-alg.com
EXAMPLE
Express Numbers in Scientific Notation
Express each number in scientific notation. Positive and Negative Exponents When the number is 1 or greater, the exponent is positive. When the number is between 0 and 1, the exponent is negative.
a. 60,000,000 60,000,000 = 6.0 × 10,000,000 The decimal point moves 7 places. = 6.0 × 107
The exponent is positive.
b. 0.0049 0.0049 = 4.9 × 0.001 The decimal point moves 3 places. = 4.9 × 10-3
The exponent is negative.
2A. 32,800
2B. 0.000064
SPACE The table shows the objects in space and their distances from the Sun. Light travels 300,000 kilometers per second. Estimate how long it takes light to travel fromthe Sun to Pluto. 109
kilometers Explore It is 5.90 × from the Sun to Pluto, and the speed of light is 300,000 kilometers per second.
Plan
Use the equation d = rt. To estimate, round 5.90 × 109 to 6.0 × 109. Write 300,000 as 3.0 × 105.
Solve 6.0 ×
Calculator To enter a number in scientific notation on a calculator, enter the decimal portion, press [EE] then enter the exponent. A calculator in Sci mode will display answers in scientific notation. For example, the number 1.0 × 1010 is displayed as 1E10 on the calculator.
d = rt 109 ≈
Earth
1.55 ⴛ 108
Jupiter
7.78 ⴛ 108
Mars
2.28 ⴛ 108
Mercury
5.80 ⴛ 107
Neptune
4.50 ⴛ 109
Pluto
5.90 ⴛ 109
Saturn
1.43 ⴛ 109
Uranus
2.87 ⴛ 109
Venus
1.03 ⴛ 108
Source: The World Almanac
Write the formula.
(3.0 ×
105)t 105)t
Replace d with 6.0 × 109 and r with 3.0 × 105.
(3.0 × 6.0 × 109 _ ≈_ 5 5
Divide each side by 3.0 × 105.
6.0 × 109 _ ≈t 5
Simplify.
3.0 × 10
Distance from the Sun (km)
Object
3.0 × 10
3.0 × 10
You can use a calculator to find the quotient. 6.0
[EE] 9 ⫼ 3.0
[EE] 5 ENTER 20000
So, it would take 20,000 (or 2.0 × 104) seconds, or about 51 hours. 2
Check
You can divide each part of the number in scientific notation. 6.0 × 109 6.0 109 _ =_ ×_ 3.0 × 105
3.0
105
= 2.0 × 104 The answer is reasonable.
3. SPACE Estimate how long it takes light to travel from the Sun to Mercury. Personal Tutor at pre-alg.com Lesson 4-7 Scientific Notation
215
Compare and Order Numbers To compare and order numbers in scientific notation, first compare the exponents. With positive numbers, any number with a greater exponent is greater. If the exponents are the same, compare the factors.
SPACE Refer to the table in Example 3. Order Mars, Jupiter, Mercury, and Saturn from least to greatest distance from the Sun. First, order the numbers according to their exponents. Then, order the numbers with the same exponent by comparing the factors. Jupiter and Mars
Step 2
Saturn
1.43 × 109 ⎭
7.78 × 108 2.28 × 108
°Ã°V
216 Chapter 4 Factors and Fractions
HOMEWORK
HELP
For See Exercises Examples 9–17 1 18–26 2 27, 28 3 29, 30 4
Express each number in standard form. 9. 4.24 × 102
10. 5.72 × 104
11. 3.347 × 10-1
12. 5.689 × 10-3
13. 1.5 × 10-4
14. 9.01 × 10-2
15. 1.399 × 105
16. 2.505 × 103
17. 6.1 × 104
Express each number in scientific notation. 18. 2,000,000
19. 499,000
20. 0.006
21. 0.0125
22. 50,000,000
23. 39,560
24. 0.000078
25. 0.000425
26. 5,894,000
27. SPACE Refer to the table in Example 3 on page 215. To the nearest second, about how long does it take light to travel from the Sun to Venus? 28. TRAFFIC In a recent year, route U.S. 59 in the Houston metropolitan area averaged approximately 338,510 vehicles per day. About how many vehicles was this during the entire year? Write the number in scientific notation. 29. OCEANS Rank the oceans in the table at the right by area from least to greatest.
Real-World Link In 2000, the International Hydrographic Organization named a fifth world ocean near Antarctica, called the Southern Ocean. It is larger than the Arctic Ocean and smaller than the Indian Ocean. Source: geography.about.com
Ocean
Area (sq mi)
Arctic
5.44 × 106
Atlantic
3.18 × 107
30. MEASUREMENT The table below shows the Indian 2.89 × 107 values of different prefixes that are used Pacific 6.40 × 107 in the metric system. Write the units attometer, gigameter, kilometer, nanometer, petameter, and picometer in order from greatest to least measure. Prefix Meaning
atto
giga
kilo
nano
peta
pico
10-18
109
103
10-9
1015
10-12
Order each set of numbers from least to greatest. 31. -3.14 × 102, -3.14 × 10-2, 3.14 × 102, 3.14 × 10-2 32. 2.81 × 104, 2805, 2.08 × 105, 3.2 × 104, 3024 33. 9,562,301, 9.05 × 10-6, 9.5 × 106, 905,000 ANALYZE GRAPHS For Exercises 34–36, use the graph. The graph shows the weights of the five heaviest marine and land mammals on Earth in pounds.
i>ÛiÃÌ >>Ã vÀV> i« >Ì Õi 7 >i
£°{{ £ä{ L Ó°nÇ £äx L {
EXTRA
PRACTICE
See pages 770, 797. Self-Check Quiz at pre-alg.com
7 >i °Ó £ä L 34. Rank the animals in order ,} Ì 7 >i n°nÓ £ä{ L from heaviest to lightest. 7 Ìi , ViÀà ǰ{ £äÎ L 35. About how many times heavier is the Blue Whale -ÕÀVi\ "OOK OF 7ORLD 2ECORDS than the African Elephant? 36. Estimate the combined weight of the Fin Whale, Right Whale, and White Rhinoceros. Write the weight in scientific notation and in standard form.
Lesson 4-7 Scientific Notation Robert Fried
217
Convert the numbers in each expression to scientific notation. Then evaluate the expression. Express in scientific notation and in standard notation. (420,000)(0.015) 38. __
20,000 37. _
0.025
0.01
H.O.T. Problems
(0.078)(8.5) 39. __ (0.16)(250,000)
40. OPEN ENDED Write a number in standard form and then write the number in scientific notation, explaining each step that you used. NUMBER SENSE Los Angeles is the second largest city in the United States. 41. Which number better describes the population of Los Angeles, 3.8 × 104 or 3.8 × 106? 42. What are some other ways to express Los Angeles’ population? 43. Which form of the number is best to use when describing population? Explain. 44. CHALLENGE In standard form, 3.14 × 10-4 = 0.000314, and 3.14 × 104 = 31,400. What is 3.14 × 100 in standard form? 45.
Writing in Math Explain how scientific notation is an important tool in comparing real-world data. Illustrate your answer with an example of real-world data that is written in scientific notation, and the advantages of using scientific notation to compare data.
46. If you wrote the areas of the bodies of water in the table from least to greatest, which would be third in the list? Body of Water
Area (km2)
Lake Huron Lake Victoria Red Sea Great Salt Lake
5.7 × 104 6.9 × 104 4.4 × 105 4.7 × 103
47. GRIDDABLE The weight of a fruit fly is about 1.3 × 10-4 pound. How many pounds would one million fruit flies weigh? 48. The distance from Earth to the Sun is about 9.6 × 107 miles. Which of the following represents this distance in standard notation?
Source: The World Almanac
A Lake Huron
C Red Sea
B Lake Victoria
D Great Salt Lake
F 9,600,000 mi
H 960,000,000 mi
G 96,000,000 mi
J
9,600,000,000 mi
ALGEBRA Evaluate each expression if s = -2 and t = 3. (Lesson 4-6) 49. t-4
50. s-5
51. 7s
52. st
ALGEBRA Find each product or quotient. Express using exponents. (Lesson 4-5) 53. 44 · 47
54. 3a2 · 5a2
55. c5 ÷ c2
57. Write ten million as a power of ten. (Lesson 4-1)
36d6 56. 4 12d
58. BUSINESS An online bookstore adds a $2.50 shipping and handling charge to the total price of every order. If the cost of books in an order is c, write an expression for the total cost. (Lesson 1-3) 218 Chapter 4 Factors and Fractions
CH
APTER
4
Study Guide and Review
wnload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
&ACTORS AN D &RACTIONS W BEL
Key Concepts Exponents
(Lesson 4-1)
• An exponent is a shorthand way of writing repeated multiplication. • Follow the order of operations to evaluate algebraic expressions containing exponents.
Prime and Composite Numbers
(Lesson 4-2)
algebraic fraction (p. 198) base (p. 180) composite number (p. 186) exponent (p. 180) factor (pp. 180 and 188) greatest common factor (GCF) (p. 191) monomial (p. 188) power (p. 180) prime factorization (p. 187) prime number (p. 186) scientific notation (p. 214) standard form (p. 214) Venn diagram (p. 191)
• A prime number is a whole number that has exactly two factors, 1 and itself. • A composite number is a whole number that has more than two factors.
Factors and Factoring
(Lesson 4-3)
• The greatest number or monomial that is a factor of two or more numbers or monomials is their greatest common factor, or GCF. • The Distributive Property can be used to factor algebraic expressions. • Algebraic fractions can be written in simplest form by dividing the numerator and the denominator by the GCF. • Powers with the same base can be multiplied by adding their exponents. Powers with the same base can be divided by subtracting their exponents.
Negative Exponents and Scientific Notation (Lessons 4-6 and 4-7)
Determine whether each statement is true or false. If false, replace the underlined word or number to make a true statement. 1. The exponent of a number raised to the first power can be omitted. 2. Numbers expressed using exponents are called powers. 3. The number 7 is a factor of 49 because it can divide into 49 with a remainder of zero. 4. A monomial is a number, a variable, or a sum of numbers and/or variables. 5. The number 64 is a composite number. 6. A number is in scientific notation when it does not contain exponents.
1 • For a ≠ 0 and any whole number n, a-n n. a
• A number in scientific notation is the product of a number between 1 and 10 and a power of 10.
Vocabulary Review at pre-alg.com
Vocabulary Check
7. A fraction is in simplest form when the GCF of the numerator and the denominator is 2.
Chapter 4 Study Guide and Review
219
CH
A PT ER
4
Study Guide and Review
Lesson-by-Lesson Review 4–1
Powers and Exponents
(pp. 180–184)
Write each expression using exponents. 8. 6 · 6 · 6 · 6 · 6 10. x · x · x
9. 4 11. f · f · g · g · g · g
Evaluate each expression if x = -3, y = 4, and z = -2. 12. 33
13. (-5)2
14. 2(3z + 4)5
15. x2z4
Example 1 Write a · a · b · b · b · b · b using exponents. Group the factors with like bases. Then, write using exponents. a · a · b · b · b · b · b = (a · a) · (b · b · b · b · b) = a2b5
Example 2 Evaluate 4(a + 2)3 if a = -5. 16. E-MAIL Suppose Theo sends an e-mail to three of his friends. Each of his three friends forwards the e-mail to three of their friends. Each of those friends forwards it to three friends, and so on. Find the number of e-mails sent during the fifth stage as a power. Then find the value of the power.
4–2
Prime Factorization
4(a + 2)3 = 4(-5 + 2)3
Replace a with -5.
= 4(-3)3
Simplify the expression inside the parentheses.
= 4(-27)
Evaluate (-3)3.
= -108
Simplify.
(pp. 186–190)
Write the prime factorization of each number. Use exponents for repeated factors. 17. 45
18. 55
19. 68
20. 200
Example 3 Write the prime factorization of 40. Use exponents for repeated factors. 40 4
Factor each monomial. 21. 18x
22. 10e2
23. 32pq
24. -25ab2
25. PHOTOGRAPHY Jacy picked out 24 photographs to put into a frame in a rectangular arrangement. How many different numbers of rows and columns can she display them in if each row has the same number of photographs? Name each arrangement.
220 Chapter 4 Factors and Fractions
· 10
40 = 4 · 10
2 · 2 · 2 · 5 4 = 2 · 2 and 10 = 2 · 5
The prime factorization of 40 is 2 · 2 · 2 · 5 or 23 · 5.
Example 4 Factor 9s3t2. 9s3t2 = 3 · 3 · s3 · t2
9=3·3
= 3 · 3 · s · s · s · t · t s3 · t2 = s · s · s · t · t
Mixed Problem Solving
For mixed problem-solving practice, see page 797.
4–3
Greatest Common Factor (GCF)
(pp. 191–195)
Find the GCF of each set of numbers or monomials.
Example 5 Find the GCF of 12a2 and 15ab.
26. 6, 48
12a2 = 2 · 2 · 3 ·
28. 4n,
5n2
27. 16, 24 29.
20c3d,
12cd
31. 3x + 24
32. 30 - 4n
33. 45s + 25
34. 14r - 30
35. 64 - 60k
Example 6 Factor 4n + 8. Step 1 Find the GCF of 4n and 8. 4n = 2 · 2 · n 8 = 2 · 2 · 2 The GCF is 2 · 2 or 4.
36. DESIGN An architect is designing two seating sections for an auditorium. One section will contain 860 seats, and the other will contain 1000 seats. Both sections will have the same number of seats per row. What is the greatest number of seats in each row?
4–4
Simplifying Algebraic Expressions
38. 24 40
39. 15 16 28w 41. 38w2 9mn 43. 18n2
40. 21 30 23x 42. 32y
Step 2 Write the product of the GCF and its remaining factors. 4n + 8 = 4(n) + 4(2) = 4(n + 2)
Rewrite using the GCF. Distributive Property
(pp. 196–200)
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. 6 37. 21
3 ·5· a ·b
The GCF of 12a2 and 15ab is 3 · a or 3a.
Factor each expression. 30. 2t + 20
15ab =
a ·a
2 44. 15ac 24ab
45. TRAVEL Of the 267 students in the freshman class, 89 of them take the bus to school. What fraction of the freshman class takes the bus to school?
. Example 7 Simplify 36 60 1 1 1
36 = 2 · 2 · 3 · 3 Divide the numerator and the 60 2 · 2 · 3 · 5 denominator by the GCF, 2 · 2 · 3 or 12. 1
1
1
= 35
Simplify.
17q2 34qr
Example 8 Simplify . 1
1
17 · q · q 17q2 = Divide the numerator and the 17 · 2 · q · r denominator by the GCF, 17 · q. 34qr 1
q = 2r
1
Simplify.
Chapter 4 Study Guide and Review
221
CH
A PT ER
4
Study Guide and Review
4–5
Multiplying and Dividing Monomials
(pp. 203–207)
Find each product or quotient. Express using exponents. 46. 84 · 85
7 47. 32
48. 7x · 2x6
49. k6 ÷ k5
3
50. LIFE SCIENCE Starting from a single bacterium, the number of bacteria after t cycles of reproduction is 2t. A bacteria reproduces every 30 minutes. If there are 1000 bacteria in a dish now, how many will there be in 1 hour?
4–6
Negative Exponents
5
Example 10 Find 43.
4 45 = 45 - 3 The common base is 4. 3 4 = 42 Subtract the exponents.
(pp. 209–213)
Write each expression using a positive exponent.
Example 11 Write 3-4 as an expression using a positive exponent.
51. 7-2
1 Definition of negative exponent 3-4 = 4
52. b-4
53. (-4)-3
Write each fraction as an expression using a negative exponent other than -1. 1 54. 3 6
1 55. 64
1 56. 125
57. DISTANCE If 1 millimeter is equal to 10-3 meter and 1 nanometer is equal to 10-9 meter, how many nanometers are in 1 millimeter?
4–7
Example 9 Find x3 · x2. x3 · x2 = x3 + 2 The common base is x. = x5 Add the exponents.
Scientific Notation
3
1 as an expression Example 12 Write 32 using a negative exponent other than -1. 1 1 = Find the prime factorization of 32. 32 2 · 2 · 2 · 2 · 2 1 = 5
Definition of exponent.
= 2-5
Definition of negative exponent.
2
(pp. 214–218)
58. 6.1 × 102
59. 2.9 × 10-3
Example 13 Express 3.5 × 10-2 in standard form.
60. 1.85 × 10-2
61. 7.045 × 104
3.5 × 10-2 = 3.5 × 0.01 10-2 = 0.01
Express each number in standard form.
= 0 035 Express each number in scientific notation.
Move the decimal point 2 places to the left.
62. 1200
63. 0.008
Example 14 Express 269,000 in scientific notation.
64. 0.000319
65. 45,710,000
269,000 = 2.69 × 100,000 The decimal point moves 5 places.
66. SPACE The mass of the Sun is 1.98892 × 1015 exagrams. Express this number in standard form. 222 Chapter 4 Factors and Fractions
= 2.69 ×
105
The exponent is positive.
CH
A PT ER
4
Practice Test
Write each expression using exponents.
Write each expression using a positive exponent.
1. 3 · 3 · 3 · 3 2. b · b · b · b · b 3. -2 · -2 · -2 · a · a · a · a 4. ALGEBRA Is the value of 2n - 1 prime or composite if n = 5? 5. MULTIPLE CHOICE When the United States had 48 states, the stars on the flag were in a 6 × 8 rectangular arrangement. Which rectangular arrangement of the 48 stars would NOT be possible?
16. 4-2
17. 10-10
18. t-6
19. (yz)-3
20. MEASUREMENT How many square centimeters is equivalent to one square millimeter? Write as an expression with a positive exponent. Write each number in standard form.
A 2 × 24
22. 5.206 × 10-3 24. 7.29 × 103
21. 3.71 × 104 23. 3.4 × 10-5
B 3 × 16 C 4 × 12 D 5 × 10
Write each number in scientific notation. 25. 0.09 27. 50,300
Factor each expression. 6. 12r2 8. 7 + 21p
7. 50xy2 9. 24c - 10
10. MULTIPLE CHOICE Eighty fluid ounces is what part of 1 gallon? (Hint: There are 128 fluid ounces in 1 gallon.) 3 F 10
H 5 8
G 2 5
J 3 4
11.
·
56
12.
(4x7)(-6x3)
9 14. w 5
13. k · k5
w
15. MULTIPLE CHOICE Which expression represents the area of the square? A 3ab2 B
3a2b2
C 9ab2 D 9a2b2
Chapter Test at pre-alg.com
ANALYZE GRAPHS For Exercises 29 and 30, use the graph. The graph shows the maximum amounts of lava in cubic meters per second that erupted from six volcanoes in the last century. Eruption Rates
Find each product or quotient. Express using exponents. 53
26. 1,068,300 28. 0.008
ÎAB
Mount St. Helens, 1980 Ngauruhoe, 1975 Hekla, 1970 Agung, 1963 Bezymianny, 1956 Hekla, 1947 Santa Maria, 1902
2.0 104 2.0 103 4.0 103 3.0 104 2.0 105 2.0 104 4.0 104
Source: University of Alaska
ÎAB
29. Rank the volcanoes in order from greatest to least eruption rate. 30. How many times greater was the Santa Maria eruption than the Mount St. Helens eruptions?
Chapter 4 Practice Test
223
CH
A PT ER
Standardized Test Practice
4
Cumulative, Chapters 1–4
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
6. In the sequence below, which expression can be used to find the value of the term in the nth position?
1. Which coordinates are most likely to be the coordinates of point P? 20
P
y
10
⫺20 ⫺10 O
10
20 x
⫺10 ⫺20
A (-13, 7)
C ( 13, 7)
B (7, -13)
D (7, 13)
2. A certain bacterium has a diameter of 0.0000000025 centimeter. How is this length expressed in scientific notation? × 10-8 cm
F 2.5
× 109 cm
H 2.5
G 2.5
× 108 cm
J 2.5 × 10-9 cm
3. The shipping charge from an Internet bookstore includes a base fee of $5 plus $3 for each item purchased. Which equation represents the shipping charge for n items? A S = 5(3n)
3 C S=5+ n
B S = 3n - 5
D S = 5 + 3n
4. Suppose you paid for a DVD with a $20 bill. You received 3 dollars, 3 dimes, and 2 pennies in change. How much did you pay for the DVD? F $16.68
H $17.68
G $16.88
J $17.88
5. Express 0.0000000102 in scientific notation. A 1.02 × 10-9
C 1.02 × 10-7
B 1.02 × 10-8
D 1.02 × 10-6
224 Chapter 4 Factors and Fractions
Position
Term
1
0.5
2
1.5
3
2.5
4
3.5
5
4.5
n
?
F n - 0.5
H 2n
n G 2
n J 4
7. GRIDDABLE The low temperatures during the past five days are given in the table. Find the average (mean) of the temperatures. Day Temperature (˚F)
1
2
3
4
5
-2
4
5
4
8. Which table of values represents the following rule? Add the input number to the square of the input number. A
C Input (x)
Output (y)
Input (x)
Output (y)
1
1
2
2
3
2
6
4
5
4
20
B
D Input (x)
Output (y)
Input (x)
Output (y)
1
1
1
2
2
6
2
4
4
8
4
8
9. GRIDDABLE A bus traveled 185 miles at an average speed of 60 miles per hour. About how many hours did it take for the bus to reach its destination? Round your answer to the nearest tenth. Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
10. The rectangle below is 8 feet long and 5 feet wide.
13. Suppose that the weight of a certain breed of puppy at 3 months is 1.3 times its weight at 2 months. Given x, the weight of a puppy at 2 months, which equation can be used to find y, the weight of the puppy at 3 months? A y = 1.3 + x C y = 1.3x B y = x - 1.3 D y = 1.3 ÷ x
x vÌ n vÌ
If the dimensions of the figure are multiplied by 3, by what factor will the area increase? F 3 H 15 G9 J 40
Pre-AP Record your answers on a sheet of paper. Show your work. 14. Chandra plans to order CDs from an Internet shopping site. She finds that the CD prices are the same at three different sites, but that the shipping costs vary. The shipping costs include a fee per order, plus an additional fee for each item in the order, as shown in the table below.
Question 10 Most standardized tests include any necessary formulas in the test booklet. It helps to be familiar with formulas such as the area of a rectangle, but use any formulas that are given to you.
Shipping Cost
11. Sydney spends 20 minutes traveling to and from work everyday. What fraction of the day is this? 1 A
Company
72 1 B 12 2 C 3 5 D 6
Per Order
Per Item
CDBargains
$4.00
$1.00
WebShopper
$6.00
$3.00
EverythingStore
$2.50
$1.50
a. For each company, write an equation that represents the shipping cost. In each of your three equations, use S to represent shipping cost and n to represent the number of items purchased.
12. The distance from Earth to Mars averages about 2.28 108 kilometers. Which of the following represents this number in standard notation? F 228,000,000 km G 22,800,000 km H 2,280,000 km J 228,000 km
b. If Chandra orders 2 CDs, which company will charge the least for shipping? Use the equations you wrote and show your work. c. If Chandra orders 10 CDs, which company will charge the least for shipping? Use the equations you wrote and show your work.
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Go to Lesson...
1-6
4-7
3-6
1-1
4-7
3-7
2-5
1-6
3-8
3-8
4-5
4-7
3-4
3-6
Chapter 4 Standardized Test Practice
225
Rational Numbers
5 •
Understand that different forms of numbers are appropriate for different situations.
•
Select and use appropriate operations to solve problems and justify solutions.
•
Uses statistical procedures to describe data.
Key Vocabulary common multiples (p. 257) least common denominator (p. 258) measures of central tendency (p. 274)
reciprocals (p. 245)
Real-World Link Hurricanes A hurricane can be measured by winds greater than 74 miles per hour, a storm surge greater than 4 feet, and barometric pressure less than 28.94 inches.
Applying Rational Numbers Make this Foldable to help you record information about rational numbers. 1 Begin with three sheets of 8_” × 11” paper. 2
1 Fold the first two sheets in half from top to bottom. Cut along the fold from edges to margin.
3 Insert the first two sheets through the third sheet and align the folds.
226 Chapter 5 Rational Numbers Getty Images
2 Fold the third sheet in half from top to bottom. Cut along the fold from margin to edge.
4 Label each page with a lesson number and title.
>«ÌiÀ x\ ,>Ì> ÕLiÀÃ
GET READY for Chapter 5 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1 Take the Quick Check below. Refer to the Quick Review for help.
Find each quotient. Round to the nearest tenth, if necessary. (Lesson 2-5) 1. 3 ÷ 5 2. -1 ÷ 8 3. 2 ÷ 17 4. -12 ÷ 3 5. -2 ÷ (-9) 6. 4 ÷ (-15) 7. -24 ÷ 14 8. -72 ÷ (-9)
Example 1
Find 3 ÷ 8. Round to the nearest tenth. 3 ÷ 8 = 0.375 ≈ 0.4
Find the quotient. Round to the nearest tenth.
9. LANDSCAPING A hedge of roses is 8.25 meters long. Suppose bricks each 0.25 meter long will make a border along one side. How many bricks are needed to make the border? (Prerequisite Skills, pp. 749–750)
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 5 12 14 10. _ 11. _ 12. _ 40 36 13. _ 50
20 27 14. _ 54
39 32 15. _ 85
16. SURVEY Ten of the 25 students in math class have blue eyes. What fraction of the students in math class do not have blue eyes? Write in simplest form. (Lesson 4-4)
Example 2 14 Write _ in simplest form. 35
Find the GCF of 14 and 35. factors of 14: 1, 2, 7, 14 factors of 35: 1, 5, 7, 35 The GCF of 14 and 35 is 7. 14 14 ÷ 7 _ = _ 35 35 ÷ 7 2 = _
Simplest form
5
Find each sum or difference. (Lesson 2-2) 17. 4 + (-9) 18. -10 + 16 19. (-3) + (-8) 20. -1 - (-10) 21. SCUBA DIVING A scuba diver descends 21 feet below the surface of the water. She then ascends 14 feet. Find an integer that represents the scuba diver’s position in relation to the surface of the water.
Divide the numerator and the denominator by the GCF.
Example 3
Find 14 - 18. 14 - 18 = 14 + (-18) = -4
To subtract 18, add -18. Simplify.
(Lesson 2-2) Chapter 5 Get Ready for Chapter 5
227
5-1
Writing Fractions as Decimals
Main Ideas • Write fractions as terminating or repeating decimals. • Compare fractions and decimals.
New Vocabulary terminating decimal mixed number repeating decimal bar notation
In the 18th century, a silver dollar contained $1 worth of silver. The sizes of all other coins were based on this coin.
Coin
Fraction of Silver of $1 Coin
quarter-dollar (quarter)
1 4
10-cent (dime)
1 10
half-dime* (nickel)
1 20
a. A half dollar contained half the silver of a silver dollar. What was it worth? b. Write the decimal value of each coin in the table. c. Order the fractions in the table from least to greatest. (Hint: Use the values of the coins.)
* In 1866, nickels were enlarged for convenience
Write Fractions as Decimals Any fraction _a , where b ≠ 0, can be b
written as a decimal by dividing the numerator by the denominator. So, _a = a ÷ b. If the division ends, or terminates, when the remainder b is zero, the decimal is a terminating decimal.
EXAMPLE
Write a Fraction as a Terminating Decimal
3 Write _ as a decimal. 8
Vocabulary Link Terminating Everyday Use bringing to an end Math Use a decimal whose digits end
Method 1 Use paper and pencil.
Method 2 Use a calculator.
0.375 3.000 8 -2.4 60 -56 40 -40 0
3 ⫼ 8 ENTER 0.375
_3 = 0.375 8
Division ends when the remainder is 0.
0.375 is a terminating decimal.
Write each fraction as a decimal. 4 1A. _ 5
3 1B. _ 16
1 is the sum of a whole number and a fraction. A mixed number such as 3_ 2 Mixed numbers can also be written as decimals.
228 Chapter 5 Rational Numbers
EXAMPLE
Write a Mixed Number as a Decimal
1 Write 3_ as a decimal. 2
Mental Math It will be helpful to memorize the following list of fraction-decimal equivalents. 1 2 = 0.5 1 3 = 0.3 1 4 = 0.25 1 5 = 0.2 2 3 = 0.6
3 4 = 0.75 2 5 = 0.4 3 5 = 0.6
4 5 = 0.8
1 1 3_ =3+_ 2
2
Write as the sum of an integer and a fraction.
= 3 + 0.5 12 = 0.5 = 3.5
Add.
Write each mixed number as a decimal. 1 2A. 2_
3 2B. 4_
4
4
Not all fractions can be written as terminating decimals.
2→ 3
CHECK 2 ⫼ 3
The number 6 repeats.
0.666 3 2.000 -1 8 20 -18 20 -18 2
The remainder after each step is 2.
.6666666667 The last digit is rounded.
So, 2 = 0.6666666666… . This decimal is called a repeating decimal. 3 Repeating decimals have a pattern in their digits that repeats without end. You can use bar notation to indicate that a digit or group of digits repeats. 6 The digit 6 repeats, so place a bar over the 6. 0.6666666666… = 0. The table shows three examples of repeating decimals.
EXAMPLE
Decimal
Bar Notation
0.13131313. . . 6.855555. . .
0. 13 6.85
19.1724724. . .
19.1 724
Write Fractions as Repeating Decimals
Write each fraction as a decimal. Use a bar to show a repeating decimal. 2 b. _
6 a. -_
11 0.5454. . . 6 → - 11 11 6.0000. . . 6 So, -_ = -0. 54. 11
7 3A. _ 9
Extra Examples at pre-alg.com
The digits 54 repeat.
15 2 → 15
0.1333. . . The digit 3 15 2.0000. . . repeats.
2 So, _ = 0.1 3. 15
5 3B. -_ 6
Lesson 5-1 Writing Fractions as Decimals
229
GOLF During the 2005 Masters Tournament, Tiger Woods’ first shot landed on the fairway 32 of 56 times. To the nearest thousandth, what part of the time did his shot land on the fairway? Divide the number of fairways on which he landed, 32, by the total number of fairways, 56. 32 4 _ =_ ≈ 0.571428… or 0. 571428 56
7
Look at the digit to the right of the thousandths place. Round down since 4 < 5. Tiger Woods landed on the fairway 0.571 of the time.
4. SOFTBALL The United States women’s softball team had 73 hits out of a total of 213 at bats in the final round of the 2004 Olympics. To the nearest thousandth, what part of the time did the team have a hit in the final round?
Compare Fractions and Decimals It may be easier to compare numbers when they are written as decimals. Real-World Link Tiger Woods became the youngest golfer to win The Masters golf tournament at the age of 21 years 3 months and 14 days when he won in 1997. Source: masters.org
EXAMPLE
Compare Fractions and Decimals
Replace each a. 35
with , or = to make a true sentence.
0.75 3 5
0.75
Write the sentence.
0.6
0.75
Write 35 as a decimal.
0.6 < 0.75
In the tenths place, 6 < 7.
0.6
0.75
0.5 0.55 0.6 0.65 0.7 0.75 0.8
3 On a number line, 0.6 is to the left of 0.75, so _ < 0.75. 5 b. -_ 8
5
6 -_ 9
Write the fractions as decimals and then compare the decimals. 5 6 -_ = -0.625 -_ = -0.6 8
9
On a number line, -0.625 is to the right of -0. 6, so -0.625 is 5 6 6. The inequality is -_ > -_ . greater than -0. 8
7 5A. _ 8
230 Chapter 5 Rational Numbers Mike Blake/REUTERS/Landov
0.87
9
7 5B. -_ 15
5 -_ 12
13 17 BREAKFAST In a survey of students, _ of the boys and _ of the girls 20 25 make their own breakfast. Of those surveyed, do a greater fraction of boys or girls make their own breakfast?
Write the fractions as decimals and then compare the decimals. 13 = 0.65 boys: _
0.65
20 17 girls: _ = 0.68 25
0.68
0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70
13 17 °®
ä°Îä
16 1 48. -5_ 3
2 47. -2.2 -2_ 7
Real-World Link After being caught, a marlin can strip more than 300 feet of line from a fishing reel in less than 5 seconds. Source: Incredible Comparisons
ä°Ón
ä°Ó£
ä°Óä ä°£È
ä°£x
ä°£x
ä°£Î
ä°£ä ä°äx
25 3 -5_ 10
>Þ `
>Þ ÕÀ
À
Ã`
Ã` /
8 -_
>Þ
Þ `>
5 46. -_
40. 6.18
ä
99
-2.09
12
4 1 6_ 5
ä°Óx
1 45. -2_
1 -_
,>v> ÕÀ} x >Þ *iÀ`
Replace each with , or = to make a true sentence. 34 7 43. -0.75 -_ 44. _ 0.3 4 9
2
6 36. -_ 25
i
1 38. 1_ 20
0. 5
1 1 35. -_ -_ 8 10 39. 34 3. 4 9
10 _ 14
_1
32. 0.3
0.4
5
>Þ
_8 9
2 31. _
0.65
Ã`
7 33. _ 8 1 37. _ 5
8 5 34. _ 7
7 i`
5 30. _
1 29. 0.3 _ 4
/Õ i
Replace each
>Þ
49. AUTOMOBILES Of all the cars sold in the United States in 2003, 2 were 5 imported from Japan and 0.26 were imported from Germany. Are more Japanese or German cars sold in the United States? Explain. Order each group of numbers from least to greatest. 7 7 50. _ , 0.8, _ 8
2 3 , -_ 51. -0.29, -
11 7 1 1 53. -1. 1, -1_, -1_ 8 10
9
3 2 , 2.67, 2_ 52. 2_ 5 3
54. ANALYZE GRAPHS Find a fraction or mixed number that might represent each point on the graph at the right.
!" ä°x
# $ £
% £°x
Ó
ANIMALS For Exercises 55 and 56, use the following information. 5 mile in one minute. A marlin can swim _ EXTRA PRACTICE
6
5 55. Write _ as a decimal rounded to the nearest hundredth. 6
See pages 770, 798.
56. Which form of the number is best to use? Explain.
Self-Check Quiz at pre-alg.com
57. OPEN ENDED Give one example each of real-world situations where it is most appropriate to give a response in fractional form and in decimal form.
232 Chapter 5 Rational Numbers Burton McNeely/Getty Images
H.O.T. Problems
58. NUMBER SENSE Find a terminating and a repeating decimal between 1 and 8. Explain how you found them. 6
9
59. CHALLENGE Write the prime factorization of each denominator and the decimal equivalent of each fraction. Then explain how prime factors of denominators and the decimal equivalents of fractions are related. 1 _ 1 _ _1 , _1 , _1 , _1 , _1 , _1 , _1 , _ , 1,_ , 1 2 3 4 5 6 8 9 10 12 15 20
60. SELECT A TECHNIQUE Luke is making lasagna that calls for 4 pound of 5 mozzarella cheese. The store only has packages that contain 0.75- and 0.85-pound of mozzarella cheese. Which of the following techniques might Luke use to determine which package to buy? Justify your selection(s). Then use the technique(s) to solve the problem. mental math
61.
Writing in Math
number sense
estimation
Explain how 0.5 and 0. 5 are different. Which is
greater?
62. Which decimal represents the shaded portion of the figure? A 0.6
C 0.63
B 0. 6
D 0.6 3
7 is found between which 63. The fraction _ 9 pair of fractions on a number line? 3 3 and _ F _
5 4 7 4 G _ and _ 5 10
3 7 H _ and _
J
10 4 3 2 _ and _ 5 3
Write each number in scientific notation. (Lesson 4-7) 64. 854,000,000
65. 0.077
66. 0.00016
67. 925,000
Write each expression using a positive exponent. (Lesson 4-6) 68. 10-5
69. (-2)-7
70. x-4
71. y -3
72. ALGEBRA Write (a · a · a)(a · a) using an exponent. (Lesson 4-1) 73. TRANSPORTATION A car travels an average of 464 miles on one tank of gas. If the tank holds 16 gallons, how many miles per gallon does it get? (Lesson 3-8) 74. SCUBA DIVING A scuba diver descends from the surface of the lake at a rate of 6 meters per minute. Where will the diver be in relation to the lake’s surface after 4 minutes? (Lesson 2-4)
PREREQUISITE SKILL Simplify each fraction. (Lesson 4-4) 4 75. 30
5 76. 65
77. 36 60
78. 12 18
79. 21 24
80. 16 28
81. 32 48
125 82. 1000 Lesson 5-1 Writing Fractions as Decimals
233
5-2
Rational Numbers
Main Ideas • Write rational numbers as fractions. • Identify and classify rational numbers.
New Vocabulary rational number
Animation pre-alg.com
The solution of 2x = 4 is 2. It is a member of the set of natural numbers. N = {1, 2, 3, …} The solution of x + 3 = 3 is 0. It is a member of the set of whole numbers. W = {0, 1, 2, 3, …}
7HOLE .UMBERS .ATURAL .UMBERS .ATURAL .UMBERS
The solution of x + 5 = 2 is -3. It is a member of the set of integers. I = { …, -3, -2, -1, 0, 1, 2, 3, …}
)NTEGERS 7HOLE .UMBERS
The solution of 2x = 3 is 3, which is neither a 2 natural number, a whole number, nor an integer. It is a member of the set of rational numbers.
.ATURAL .UMBERS
Rational numbers include fractions and decimals as well as whole numbers and integers. a. Is 7 a natural number? a whole number? an integer?
,>Ì>
b. How do you know that 7 is also a rational number?
ÕLiÀÃ
)NTEGERS 7HOLE .UMBERS
c. Is every whole number a rational number? Is every rational number a whole number? Give an explanation or a counterexample to support your answers.
.ATURAL .UMBERS
Reading Math
Write Rational Numbers as Fractions A number that can be written as
Ratios Rational comes from the word ratio. A ratio is the comparison of two quantities by division. Recall that _a = a ÷ b, where
a fraction is called a rational number. Some examples of rational numbers are shown below. 3 28 5 1 1 0.75 = _ -0.3 = – _ 28 = _ 1_ =_
b
b ≠ 0.
4
EXAMPLE
3
1
4
Write Mixed Numbers and Integers as Fractions
Write each rational number as a fraction. 2 a. 5 _
3 17 _ 52 =_ 3 3
3 1A. 4_ 4
234 Chapter 5 Rational Numbers
4
b. -3 Write 5 23 as an improper fraction.
-3 3 -3 = _ or - _ 1
1B. 7
1
Terminating decimals are rational numbers because they can be written as a fraction with a denominator of 10, 100, 1000, and so on.
EXAMPLE
Write Terminating Decimals as Fractions
Simplify. The GCF of 48 and 100 is 4.
1000 3 = 6_ 8
6.375 is 6 and 375 thousandths. Simplify. The GCF of 375 and 1000 is 125.
2A. 0.56
hu nd red ths tho us an dth ten s -th ou sa nd ths
0 4 8
b. 6.375 375 6.375 = 6_
ten ths
0.48 is 48 hundredths.
ten ths
100 12 =_ 25
on es
48 0.48 = _
on es
a. 0.48
hu nd red ths tho us an dth ten s -th ou sa nd ths
Write each decimal as a fraction or mixed number in simplest form. tho us an ds hu nd red s ten s
Decimal Point Use the word and to represent the decimal point. • Read 0.375 as three hundred seventy-five thousandths. • Read 300.075 as three hundred and seventy-five thousandths.
tho us an ds hu nd red s ten s
Reading Math
6 3
5
2B. 5.875
Any repeating decimal can be written as a fraction, so repeating decimals are also rational numbers.
EXAMPLE Repeating Decimals When two digits repeat, multiply each side by 100. Then subtract N from 100N to eliminate the repeating part.
Write Repeating Decimals as Fractions
Write 0. 8 as a fraction in simplest form. N = 0.888…
Let N represent the number.
10N = 10(0.888…) Multiply each side by 10 10N = 8.888…
because one digit repeats.
Subtract N from 10N to eliminate the repeating part, 0.888… . 10N = 8.888… -(N = 0.888…) 9N = 8
10N - N = 10N - 1N or 9N
9N 8 _ =_
Divide each side by 9.
9
9 8 N=_ 9
CHECK
Simplify.
8 ⫼ 9 ENTER .8888888889
3. Write 0. 3 as a fraction in simplest form. Personal Tutor at pre-alg.com Lesson 5-2 Rational Numbers
235
READING in the Content Area For strategies in reading this Lesson, visit pre-alg.com.
Identify And Classify Rational Numbers All rational numbers can be written as terminating or repeating decimals. Decimals that are neither terminating nor repeating, such as the numbers below, are called irrational. You will learn more about irrational numbers in Chapter 9.
= 3.141592654…
→ The digits does not repeat.
4.232232223…
→ The same block of digits does not repeat.
Rational Numbers A rational number is any number that can be expressed a as the quotient _ of two b integers, a and b, where b ≠ 0.
Words
Model Rational Numbers 1.8 0.7
EXAMPLE
2
45 Integers ⫺12 ⫺5 Whole 2 Numbers 3 6 0 15 2 ⫺3.2222...
Classify Numbers
Identify all sets to which each number belongs. a. -6 -6 is an integer and a rational number. 4 b. 2 _ 5
4 14 =_ , it is rational. It is neither a whole number nor an integer. Because 2_ 5
5
c. 0.914114111… This is a nonterminating, nonrepeating decimal. So, it is not rational.
6 4A. -_ 9
Example 1 (p. 234)
4B. 1.414213562…
4C. 0
Write each number as a fraction. 1 1. -2 _ 3
5 2. 1 _ 6
3. 10
(p. 235)
4. MEASUREMENT A micron is a unit of measure that is approximately 0.000039 inch. Express this as a fraction.
Examples 2, 3
Write each decimal as a fraction or mixed number in simplest form.
Example 2
(p. 235)
5. 0.8 8. -0. 7
Example 4 (p. 236)
6. 6.35 9. 0.45
7. 3.16 10. 0.06
Identify all sets to which each number belongs. 11. -5
236 Chapter 5 Rational Numbers
12. 6.05
13. 0. 1 Extra Examples at pre-alg.com
HOMEWORK
HELP
For See Exercises Examples 14–17 1 18–25 2 26–31 3 32–39 4
Write each number as a fraction. 2 14. 4_
4 15. -1_
3
16. -21
7
17. 60
Write each decimal as a fraction or mixed number in simplest form. 18. 0.4
19. 0.09
20. 5.22
21. 1.68
22. 3.625
23. 8.004
24. WHITE HOUSE The White House covers an area of 0.028 square mile. What fraction of a square mile is this? 25. RECYCLING Use the information at the left to find the fraction of all recycled newspapers that were used to make tissues in 2004. Write each decimal as a fraction or mixed number in simplest form. 27. -0.333… 28. 4. 5 26. 0.2 29. 5. 6
30. 0. 32
31. 2. 05
Identify all sets to which each number belongs. 32. 4
33. -7
5 34. -2_
6 35. _
36. 15.8
37. 9.0202020…
38. 1.7345…
39. 30.151151115…
8
3
40. TRACK AND FIELD During the women’s 100-meter final in the 2004 Olympics, the eight finalists finished within twenty-five hundredths of a second of each other. Write this number as a fraction in simplest form. Real-World Link The portions of recycled newspapers used for other purposes are shown below. Newsprint: 0.31 Exported for Recycling: 0.28 Paperboard: 0.13 Tissues: 0.08 Other products: 0.18 Source: American Forest and Paper Association, Newspaper Association of America
41. ANALYZE TABLES The city of Heath makes 1 of the population in Rockwall County. up _ 10 Use the table to find the fraction of Rockwall County’s population that lives in other cities. Write each fraction in simplest form.
Replace each 42. -0.23
PRACTIICE
Decimal Part of Rockwall County’s Population
Fate
0.018
McLendonChisholm Rockwall Royse City
0.02 0.42 0.07
with , or = to make a true statement.
-0.3
1 45. -1_ -0.9 11
EXTRA
City
43. 8 9
0.888…
5 4_ 46. 4.63 8
44. 0.714
_5
47. -5. 3
5.333…
7
3 -inch 48. MACHINERY Will a steel peg 2.37 inches in diameter fit in a 2 _ 8 diameter hole? How do you know?
See pages 770, 798. Self-Check Quiz at pre-alg.com
H.O.T. Problems
49. GEOMETRY Pi () to six decimal places has a value of 3.141592. Pi is often . Is the estimate for greater than or less than the actual estimated as 22 7 value of ? Explain. 50. OPEN ENDED Give an example of a number that is not rational. Explain why it is not rational. Lesson 5-2 Rational Numbers
Lester Lefkowitz/CORBIS
237
51. CHALLENGE Show that 0.999… = 1. REASONING Determine whether each statement is sometimes, always, or never true. Explain by giving an example or a counterexample. 52. An integer is a rational number. 53. A rational number is an integer. 54. A whole number is not a rational number. 55.
Writing in Math
Explain how rational numbers are related to other sets of numbers. Illustrate your reasoning with examples of numbers that belong to more than one set and examples of numbers that are only rational.
56. There are infinitely many between S and T on the number line. S
57. Which fraction is between 0.12 and 0.15? 3 F _
T
25
1 G_ 8
⫺2⫺1 0 1 2 3 4 5 6 7 8
A rational numbers
3 H_ 20 1 J _ 5
B integers C whole numbers D negatives
Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1) 2 58. _
4 59. -7 _
5
5
13 60. -_
5 61. 2_
64. 7.4 × 10-4
65. 1.681 × 10-2
9
20
Write each number in standard form. (Lesson 4-7)
62. 2 × 103
63. 3.05 × 106 2
12n 66. ALGEBRA Write _ in simplest form. (Lesson 4-4) 3an
Find the perimeter and area of each rectangle. (Lesson 3-8)
67.
68. {°x V
7 in. V
16 in.
69. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than 8 times the number of nickels in the collection. How many nickels does Jada have in her collection? (Lesson 3-6)
PREREQUISITE SKILL Estimate each product. (page 752) 2 1 70. 1_ · 4_ 3
8
1 4 71. -5_ · 3_
238 Chapter 5 Rational Numbers
3
5
1 1 72. 2_ · 2_ 4
9
9 7 73. 6_ · 1_ 8
10
5-3
Multiplying Rational Numbers
Main Ideas • Multiply positive and negative fractions. • Use dimensional analysis to solve problems.
New Vocabulary
To find 2 · 3, use an area model
3 4
3 4 2 to find of 3. 3 4
2 3
a. The overlapping green area 2 represents the product of _
3 3 and _. What is the product? 4
dimensional analysis
Draw a rectangle and shade three fourths of it yellow.
Then shade two thirds of the rectangle blue.
Use an area model or another manipulative to model each product. Explain how the model shows the product. 3 1 c. _ · _
1 _ b. _ ·1 2
5
3
3 1 d. _ · _ 4
4
3
e. What is the relationship between the numerators and denominators of the factors and the numerator and denominator of the product?
Multiply Fractions This model suggests a rule for multiplying fractions. Multiplying Fractions To multiply fractions, multiply the numerators and multiply the denominators.
Words Symbols
a _ a·c _ ·c=_ , where b, d ≠ 0 b
d
b·d
1 2 2 1·2 Example _ · _ = _ or _ 3 5 15 3·5
EXAMPLE
Multiply Fractions
2 _ Find _ · 3 . Write the product in simplest form. 3
Review Vocabulary GCF (greatest common factor) the greatest number that is a factor of two or more numbers; Example: for 12 and 20, the GCF is 4. (Lesson 4-4)
4
←Multiply the numerators. ←Multiply the denominators. 3·4 6 1 =_ or _ Simplify. The GCF of 6 and 12 is 6. 2 12
2·3 _2 · _3 = _ 3
4
Find each product. Write in simplest form. 1 _ 1A. _ · 4 2
10
5 _ 1B. _ · 6 12
10
Lesson 5-3 Multiplying Rational Numbers
239
If the fractions have common factors in the numerators and denominators, you can simplify before you multiply.
EXAMPLE
Multiply Negative Fractions
5 _ Find -_ · 3 . Write the product in simplest form. Negative Fractions
12 8 1 5 _ 5 _ -_ · 3 = -_ ·3 12 8 12 8
Divide 3 and 12 by their GCF, 3.
4
5 can be written as - 12
-5 · 1 =_
Multiply the numerators and multiply the denominators.
4·8 5 = -_ 32
-5 5 or as . 12 -12
Simplify.
Find each product. Write in simplest form. 3 _ 6 3 · 9 2B. _ · -_ 2A. _ 4
9
12
EXAMPLE
11
Multiply Mixed Numbers
Find 125 · 21. Write the product in simplest form. Estimate 1 · 3 = 3 2
12 1 7 _ _ · 2_ =_ ·5 5
2
5
Rename 125 as 75 and rename 212 as 52 .
2
1
7 _ =_ ·5
Estimation
5
You can justify your answer by using estimation.
7·1 =_
1·2 7 1 = _ or 3_ 2 2
• 125 is close to 1. • 212 is close to 3. So, 125 · 212 ≈ 1 · 3 or 3.
Divide by the GCF, 5.
2
1
Multiply. Simplify.
Find each product. Write in simplest form. 3 5 1 1 3A. 3_ · 2_ 3B. -1_ · 4_ 8
3
6
8
ROLLER COASTERS The first drop on one roller coaster at a theme park is 255 feet. The first drop on another roller coaster at the park is about 11 _ as high. Find the height of the drop on the second roller coaster. 20
11 . To find the height of the drop on the second roller coaster, multiply 255 by _ 20 255 _ 11 255 255 · _ =_ · 11 Rename 255 as _ . 20
1
1
20
51
255 _ =_ · 11 1
20
Divide by the GCF, 5.
4
51 · 11 =_
1·4 561 1 _ = or 140_ 4 4
240 Chapter 5 Rational Numbers
Multiply. Simplify. The height of the drop is about 140 feet. Extra Examples at pre-alg.com
4. SKYSCRAPERS The Sears Tower in Chicago is about 1450 feet. The Empire State Building in New York City is about 4 as tall. About how tall is the 5 Empire State Building? Algebraic fractions are multiplied in the same manner as numeric fractions.
EXAMPLE
Multiply Algebraic Fractions
b2
2a _ Find _ · . Write the product in simplest form. b
d
1
b2
2a _ _ b·b _ · = 2a · _ b
d
d
b 1
2ab =_
The GCF of b and b2 is b. Simplify.
d
Find each product. Write in simplest form. x2 _ · z 5A. _ 3y
2 7 _ 5B. _ · rs 2
2x
r
10
Dimensional Analysis Dimensional analysis is the process of including units of measurement as factors when you compute. You can use dimensional analysis to check whether your answers are reasonable.
SPACE TRAVEL The landing speed of the space shuttle is about 216 miles per hour. How far does the shuttle travel in 13 hour during landing? Words
equals the
Variable
multiplied by the
Let
Equation
.
·
=
216 miles _ d=_ · 1 hour distance = rate · time 1 hour
3
72
216 miles _ =_ · 1 hour
1 hour
3
Divide out the common factors and units.
1
Simplify. The space shuttle travels 72 miles in _ hour 3 during landing. 1 Multiplying by _ is the same as dividing by 3. 3 1 216 · = 216 ÷ 3 3 1
= 72 miles CHECK
= 72
6. SPEED RECORD The record for the fastest land car speed is about 760 miles 1 hour? per hour. How far would the car travel in _ Personal Tutor at pre-alg.com
4
Lesson 5-3 Multiplying Rational Numbers
241
Examples 1– 3 (pp. 239–240)
Find each product. Use an area model if necessary. 1 _ 1. _ ·3
4 5 1 _ 4. 3 · _ 7 6 1 _ 7. _ -5 2 6 1 2 10. 3 _ · -_ 4 11
Example 4 (pp. 240–241)
Example 5 (p. 241)
3 5 2 _ 5. 5 · _ 10 9 2 _ 8. -_ -1 3 6 3 1 11. -5 _ · -3 _ 3 8
3 _ 3. _ ·1 8
4
4 _ 6. _ ·5 5 8 6 _ 9. -_ ·1
10 8 1 2 12. -2 _ · 5_ 2 3
13. GEOGRAPHY “Midway” is the name of 252 towns in the United States. “Pleasant Hill” occurs 5 as many times. How many towns named “Pleasant 9 Hill” are there in the United States? ALGEBRA Find each product. Write in simplest form. 2 _ · 3x 14. _ x
7
5b 15. _a · _ b
4t _ 16. _ · 18r 2
c
t
9r
(p. 241)
1 17. TRAVEL A car travels 65 miles per hour for 3_ hours. What is the distance 2 traveled? Use the formula d = rt to solve the problem and show how you can divide by the common units.
HELP
Find each product. Use an area model if necessary.
Example 6
HOMEWORK
1 _ 2. _ ·2
For See Exercises Examples 18–27 1–2 28–35 3 36, 37 4 38–43 5 44, 45 6
6 _ 18. -_ ·2
7 7 3 _ ·3 21. -_ 4 5 2 _ ·5 24. _ 5 6
4 _ 19. _ ·2
1 1 20. _ -_
9 3 5 _ 22. _ · 8 9 25 8 _ 25. _ · 27 9 28
5
8
1 2 23. -_ -_ 2
7
3 1 26. _ -_ 4
3
27. -7 · 2 8 5
7 28. 2 ·
6 29. (-3) 15
2 _ 30. 6 _ ·1
5 1 31. -_ · 3_
1 2 32. 2_ · 6_
3 2 5 1 · 2_ 33. 3 _ 3 8
12
9 12 2 1 34. -6_ -1_ 3 2
3 7 3 4 35. 1_ -9 _ 7 5
36. BREAD The average person living in Slovakia consumes about 320 pounds of bread per year. The average person living in the United States consumes 1 as much. How many pounds of bread does the average American about _ 5 consume every year? 37. BRIDGES The Golden Gate Bridge in San Francisco is 4200 feet long. 19 as long. How long is the The Brooklyn Bridge in New York City is _ 50 Brooklyn Bridge? ALGEBRA Find each product. Write in simplest form. 4a _ ·3 38. _
5 a 2 8 _ 41. _ · c c 11
242 Chapter 5 Rational Numbers
3x _ 9y 39. _ y · x n _ 42. _ · 64 18
n
12 _ 40. _ · 3k
4 jk 2 x _ 43. _ · 2z 2z 3
44. HYBRID CARS Hybrid cars can get up to 52 miles per gallon of gas. How far 3 gallon of gas? can the car travel on _ 4
2 1 45. LAWN CARE Dexter’s lawn is _ of an acre. If 7_ bags of fertilizer are needed 3 2 for 1 acre, how much will he need to fertilize his lawn?
Real-World Link An improvement of 5 miles per gallon in fuel economy saves 55 million metric tons of carbon emissions per day. Source: hybridcars.com
ANALYZE TABLES For Exercises 46–48, use the table that shows statistics from the last election for class &RACTION OF STUDENT BODY THAT VOTED ? president. ? &RACTION OF VOTES FOR (ECTOR 46. What fraction of the student ? body voted for Hector? &RACTION OF VOTES FOR .ORA 47. What fraction of the student body voted for Nora? 48. Was there another candidate for class president? How do you know? Explain your reasoning. If there was another candidate, what fraction of the student body voted for this person? 49. FILMS The table shows the number of sports films created with different themes. Which theme occurs 5 as many times as boxing? 12
Sport Theme Boxing Horse Racing Football Baseball
3 4 50. ALGEBRA Evaluate (xy)2 if x = _ and y = -_ . 4
5
Films 204 139 123 85
Source: Top 10 of Everything
MEASUREMENT Complete. 5 51. ? feet = _ mile
3 52. ? ounces = _ pound
6
8
(Hint: 1 mile = 5280 feet)
(Hint: 1 pound = 16 ounces)
2 hour = ? minutes 53. _
3 54. _ yard = ? inches
3
4
CONVERTING MEASURES Use dimensional analysis and the fractions in the table to find each missing measure. EXTRA
PRACTICE
See pages 771, 798. Self-Check Quiz at pre-alg.com
? cm 55. 5 in. = _____
56. 10 km = _____ ? mi
57. 26.3 cm = _____ ? in.
2 2 58. 8_ ft = _____ ? m2
Customary→ Metric
3 3 60. _ cm = _____ ? in. 4
59. 72 m2 = _____ ? ft2 61. 130.5 mi = _____ ? km
H.O.T. Problems
Conversion Factors Metric→ Customary
2.54 cm 1 in. 1.61 km 1 mi
0.39 in. 1 cm 0.62 mi 1 km
0.09 m2 1 ft2
10.76 ft2 1 m2
62. 130.5 km = _____ ? mi
63. OPEN ENDED Choose two rational numbers whose product is a number between 0 and 1. 5 _ · 18 . Who is correct? 64. FIND THE ERROR Terrence and Marie are finding _ 24 25 Explain your reasoning.
Terrence 1
5
3
18
Marie 3
· 25 = 24 20
4
5
1
9
5
18
4
5
9 · = 24 25 20
Lesson 5-3 Multiplying Rational Numbers Ford Motor Company
243
CHALLENGE Use the digits 3, 4, 5, 6, 8, or 9 no more than once to make true sentences. 6 □ _ 65. _ × □ =_ □
67.
□
5 □ _ 66. _ × □ =_ □
5
□
8
Writing in Math Use the information about fractions on page 239 to explain how multiplying fractions is related to areas of rectangles. Illustrate your reasoning with an area model.
68. What is the equivalent length of a chain that is 52 feet long?
8 3 69. The product of _ and _ is a 8 15 . number
A 4 yd 5 ft
F between 0 and 1
B 4.5 yd
G between 1 and 2
C 17 yd 1 ft
H between 2 and 3
D 17.1 yd
J greater than 3
Write each decimal as a fraction or mixed number in simplest form. (Lesson 5-2)
70. 0.18
71. -0.2
73. 0. 7
72. 3.04
1 74. FOOD In an online survey, about _ of teenagers go to sleep between 9 and 4 13 _ of teenagers go to sleep at 12 A.M. or later. Which group is 10 P.M., while 50 larger? (Lesson 5-1)
Express each situation with a number in scientific notation. (Lesson 4-7) 75. The number of possible ways that a player can play the first four moves in a chess game is 3 billion. 76. A particle of dust floating in the air weighs 0.000000753 gram.
77. ALGEBRA What is the product of x2 and x4? (Lesson 4-5) GEOMETRY Find the perimeter and area of each rectangle. (Lesson 3-8) 78.
79. 5 in.
3.5 m 12 in.
4.9 m
ALGEBRA Solve each equation. Check your solution. (Lesson 3-5) 80. 2x - 1 = 9
81. 14 = 8 + 3n
k 82. 7 + _ = -1 5
83. ALGEBRA Simplify 4(y + 2) - y. (Lesson 3-2)
PREREQUISITE SKILL Find the GCF of each pair of monomials. (Lesson 4-3) 84. 8n, 16n
85. 5ab, 8b
244 Chapter 5 Rational Numbers
86. 9k, 27
87. 4p2, 6p
5-4
Main Ideas
Dividing Rational Numbers
1 1 The model shows 4 ÷ _ . Each of the 4 circles is divided into _ -sections. 3
• Divide positive and negative fractions using multiplicative inverses. • Use dimensional analysis to solve problems.
New Vocabulary multiplicative inverses reciprocals
3
1 2
4 3
5
7 6
8
10 9
11
12
1 1 = 12. Another way to find the There are twelve _ -sections, so 4 ÷ _ 3 3 number of sections is to multiply 4 × 3 = 12.
Use a circle model or another manipulative to model each quotient. Explain how the model shows the quotient. 1 a. 2 ÷ _
1 b. 4 ÷ _
3
1 c. 3 ÷ _
2
4
d. MAKE A CONJECTURE Write about how dividing by a fraction is related to multiplying.
Reading Math Synonyms Multiplicative inverse and reciprocal are different terms for the same concept. They may be used interchangeably.
Divide Fractions Rational numbers have all of the properties of integers. 3 1 _ · = 1. Two numbers whose product Another property is shown by _ 3
1
is 1 are called multiplicative inverses or reciprocals. Inverse Property of Multiplication Words
The product of a number and its multiplicative inverse is 1.
Symbols
a For every number _ , where a, b ≠ 0, there is exactly one b
a _ b b _ number _ a such that b · a = 1.
EXAMPLE
Find Multiplicative Inverses
Find multiplicative inverse of each number. 1 b. 2 _
3 a. - _
8 8 _ =1 - 3 -_ 8 3
( )
5
The product is 1.
The multiplicative inverse 3 8 or reciprocal of -_ is -_ . 8 3
7 1A. -_ 9
1 _ 2_ = 11
5 5 5 11 _ _ · =1 5 11
Write as an improper fraction. The product is 1.
1 _ The reciprocal of 2_ is 5 . 5
11
1 1B. 6 _ 3
Lesson 5-4 Dividing Rational Numbers
245
1 Dividing by 2 is the same as multiplying by _ , 2 its multiplicative inverse. This is true for any rational number.
reciprocals
1 6·_ =3
6÷2=3
2
same result
Dividing Fractions Words
To divide by a fraction, multiply by its multiplicative inverse.
Symbols
a a d c _ ÷_ =_·_ , where b, c, d ≠ 0
Example
1 7 5 1 _ _ ÷_ =_ · 7 or _
d
b 4
b
7
EXAMPLE
4
c
5
20
Divide by a Fraction or Whole Number
Find each quotient. Write in simplest form. Dividing By a Whole Number When dividing by a whole number, always rename it as an improper fraction first. Then multiply by its reciprocal.
5 1 _ a. _ ÷
5 b. _ ÷6
3 9 _1 ÷ _5 = _1 · _9 3 9 3 5 3 1 _ =_ ·9 3 5
8
_5 ÷ 6 = _5 ÷ _6
Multiply by the 5 9 reciprocal of _, _.
6 Write 6 as _. 1 8 1 5 _ 1 Multiply by the _ = · 6 1 8 6 reciprocal of _1 , _6 . 5 =_ Multiply. 48
8
9 5
Divide by the GCF, 3.
1
3 =_
Simplify.
5
3
5 3 2B. _ ÷ -_
1 _ 7 2A. _ ÷
8
15
3 2C. _ ÷ 11
6 2D. -_ ÷ 12
4
4
7
To divide by a mixed number, you can rewrite the divisor as an improper fraction.
EXAMPLE
Divide by a Mixed Number
1 1 Find -7_ ÷ 2_. Write the quotient in simplest form. 2 10 1 1 15 _ 21 ÷ -7_ ÷ 2_ = -_ 2 2 10 10 15 10 = -_ · _ 2 21 5
5
2
21
Rename the mixed numbers as improper fractions. Multiply by the multiplicative inverse of _, _. 21 10 10 21
15 _ = -_ · 10 1
Divide out common factors.
7
25 4 = -_ or -3_ 7
7
Simplify.
Find each quotient. Write in simplest form.
3 1 3A. 6_ ÷ -4_ 8
246 Chapter 5 Rational Numbers
4
4 2 3B. -6_ ÷ -2_ 5
5
Extra Examples at pre-alg.com
You can divide algebraic fractions in the same way that you divide numerical fractions.
EXAMPLE
Divide by an Algebraic Fraction
3xy 2x Find _ ÷ _ . Write the quotient in simplest form. 8 4 3xy 3xy 8 2x 8 2x _ _ ÷ _ = _ · _ Multiply by the multiplicative inverse of _ , . 8 2x 4 8 2x 4 1
3xy 4
2
8 =_·_ 1
2x
Divide out common factors.
1
6y = _ or 3y 2
Simplify.
Find each quotient. Write in simplest form. mn2 m2n 4B. _ ÷ _
5ab _ 10b 4A. _ ÷ 6
4
7
8
Dimensional Analysis Dimensional analysis is a useful way to examine the solution of division problems.
Real-World Link Organized cheerleading is over a hundred years old. In November of 1898, John Campbell led the crowd at the University of Minnesota football game in the first-ever organized cheer. Source: Official Cheerleader Handbook
CHEERLEADING How many cheerleading uniforms can be made with 3 7 74 _ yards of fabric if each uniform requires 2_ yards? 8 4 3 7 To find how many uniforms, divide 74_ by 2_ . Think: How many 2 _7 s are in 74_3 ? 3 299 23 7 74_ ÷ 2_ = _ ÷ _ 8
4
8 4 299 _ 8 _ = · 4 23 13
2
4
23
299 _ =_ · 8 1
= 26
8
8
4
4
Write 74_ and 2_ as improper fractions. 3 4
7 8
Multiply by the reciprocal of _, _. 23 8 8 23
Divide out common factors.
1
Simplify.
So, 26 uniforms can be made.
CHECK
Use dimensional analysis to examine the units. yards uniforms
uniforms yards
yards ÷ _ = yards · _ Divide out the units. = uniforms
Simplify.
The result is expressed as uniforms.
3 2 5. BREAKFAST A box of cereal contains 15_ ounces. If a bowl holds 2_ ounces 5 5 of cereal, how many bowls of cereal are in one box? Personal Tutor at pre-alg.com Lesson 5-4 Dividing Rational Numbers Tony Anderson/Getty Images
247
Example 1 (p. 245)
Find the multiplicative inverse of each number. 4 1. _
Examples 2– 3 (p. 246)
(p. 247)
Example 5 (p. 247)
HOMEWORK
HELP
For See Exercises Examples 17–22 1 23–36 2 37–40 3 41–46 4 47, 48 5
8
Find each quotient. Use an area model if necessary.
5 2 5. -_ ÷ -_
6 1 _ 4. _ ÷
2 7 7 7. _ ÷ (-14) 9 8 1 ÷ 3_ 10. -_ 5 9
Example 4
1 3. 3 _
2. -16
5
3
6
4 8. _ ÷ (-2) 5 1 1 11. 2_ ÷ -1_ 6 5
8 4 _ 6. -_ ÷
9 5 1 9. 7_ ÷ 5 3 2 1 12. -5 _ ÷ 2_ 7 7
ALGEBRA Find each quotient. Write in simplest form. 14 _ 1 13. _ n ÷n
x2 _ ax 15. _ ÷
ab _ b 14. _ ÷ 6
4
5
2
16. CARPENTRY How many boards, each 2 feet 8 inches long, can be cut from a board 16 feet long if there is no waste?
Find the multiplicative inverse of each number. 6 17. _
1 18. -_
19. -7
20. 24
1 21. 5_
2 22. -3 _
11
5
9
4
Find each quotient. Use an area model if necessary. 3 1 _ 23. _ ÷
2 _ 1 24. _ ÷
5 1 _ 25. -_ ÷
6 4 ÷ -_ 26. _
8 _ 4 27. _ ÷
7 _ 14 28. _ ÷
3 _ 3 ÷ 29. _
2 2 30. _ ÷ -_
3 5 31. -_ ÷ -_
3 1 ÷ -_ 32. -_
4 33. 12 ÷ _
4 34. -8 ÷ _
5 ÷ (-4) 35. -_
2 36. 6_ ÷5
3 5 37. 3_ ÷ 1_
5
4
( 5)
11 4
9 9
1 1 ÷ -1_ 38. 7_
(
( 9)
5
)
9
5
6
10
1 2 39. -6_ ÷ 3_ 9
15
5
3
8
2
8
9
5
6
2
3
9
4
10
4
3
3 2 40. -10_ ÷ -2_
(
5
5
)
ALGEBRA Find each quotient. Write in simplest form. a a 41. _ ÷_ 7
EXTRA
PRACTIICE
See pages 771, 798. Self-Check Quiz at pre-alg.com
42 5s _ 6rs 44. _ ÷ t t
10 _ 5 42. _ ÷ 3x
cd 43. _c ÷ _
2x
8
k3
2s _ st3 46. _ ÷ 2
k 45. _ ÷ _ 9
5
24
t
8
3 1 47. FOOD How many _ -pound hamburgers can be made from 2_ pounds of 4 4 ground beef? 1 cups of sugar. How many batches 48. COOKING A batch of cookies requires 1_ 2
1 cups of sugar? of cookies can be made from 7_ 2
248 Chapter 5 Rational Numbers
ALGEBRA Evaluate each expression. 8 7 49. m ÷ n if m = -_ and n = _
3 1 50. r2 ÷ s2 if r = -_ and s = 1_
18
9
3
4
1 hours and earned $19.50. What was 51. BABY-SITTING Barbara baby-sat for 3_ 4 her hourly rate?
H.O.T. Problems
52. OPEN ENDED Write a division expression that can be simplified by using 7 . the multiplicative inverse of _ 5
3 1 _ 1 1 by _ , 1, _ , and _ . What happens to the quotient as 53. CHALLENGE Divide _ 4
2 4 8
12
the value of the divisor decreases? Make a conjecture about the quotient 3 by fractions that increase in value. Test your conjecture. when you divide _ 4
54.
Writing in Math
Explain how dividing by a fraction is related to multiplying. Illustrate your reasoning by including a model of a whole number divided by a fraction.
1 A promotional poster is printed on 16-inch by 24 _ -inch 2 posterboard and the space between the three sections and both top and bottom of the poster are shown.
55. If the total length of the three sections is 18 1 inches, 2 how long are each of the three equal sections? 1 in. A 2_ 3
5 C 4_ in. 8
4 B 3_ in.
1 D 6_ in.
15
IN
IN
IN
IN
6
1 inches, what is the 56. GRIDDABLE If the width of one section is 14 _ 4 area of one section? Round your answer to the nearest hundredth.
Find each product. Write in simplest form. (Lesson 5-3) 3 _ 57. _ ·1 5
3
15 2 58. -_ · -_ 9
16
4 _ 59. -2_ ·3 5
8
5 1 60. -_ · 1_ 12
7
Identify all sets to which each number belongs. (Lesson 5-2)
61. 16
62. -2.8888 …
63. 0. 8
64. 5.121221222 …
65. COMPUTERS In a survey, 17 students out of 20 said they use a computer as a reference source for school. Write 17 out of 20 as a decimal. (Lesson 5-1)
PREREQUISITE SKILL Write each improper fraction as a mixed number in simplest form. (Lesson 4-4) 9 66. _ 4
8 67. _ 7
17 68. _ 2
24 69. _ 5
Lesson 5-4 Dividing Rational Numbers
249
5-5
Adding and Subtracting Like Fractions
Main Ideas • Add like fractions. • Subtract like fractions.
Measures of different parts of an insect are shown. The sum of the parts is _6 inch. Use a ruler to find each 8 measure. 3 1 in. + _ in. a. _
5 in. 8
3 4 b. _ in. + _ in.
8 8 4 4 _ _ c. in. + in. 8 8
1 in. 8
6 in. 8
8 8 6 _ _ d. in. - 3 in. 8 8
Add Like Fractions Fractions with the same denominator are called like fractions.
Adding Like Fractions Words
To add fractions with like denominators, add the numerators and write the sum over the denominator.
a+b a b _ _ Symbols _ c + c = c , where c ≠ 0
EXAMPLE
1+2 3 1 2 Example _ + _ = _ or _ 5
5
5
5
Add Fractions
Find each sum. Write in simplest form. 3 5 +_ a. _
7 7 3+5 3 _ + _5 = _ 7 7 7 8 1 = _ or 1_ 7 7
Estimate 0 + 1 = 1 The denominators are the same. Add the numerators. Simplify and rename as a mixed number.
Compared to the estimate, the answer is reasonable. 5 -7 + _ b. _
( ) ( )
8 8 5 + (-7) 5 -7 _+ _ =_ 8 8 8 -2 1 = _ or -_ 8 4
1 1 Estimate _ + (-1) = -_ 2
2
The denominators are the same. Add the numerators. Simplify.
Compare your answer to the estimate. Is it reasonable?
5 4 1A. _ +_ 6
250 Chapter 5 Rational Numbers John Cancalosi/Stock Boston
6
6 4 1B. _ + -_ 7
( 7) Extra Examples at pre-alg.com
EXAMPLE
Add Mixed Numbers
5 1 Find 6 _ + 1_ . Write the sum in simplest form. Estimate 7 + 1 = 8 Alternative Method You can also stack the mixed numbers vertically to find the sum. 6_
8 8 5 5 1 1 6_ + 1_ = (6 + 1) + _ +_ 8 8 8 8 5+1 _ =7+ 8 6 3 _ = 7 or 7_ 8 4
(
)
Add the whole numbers and fractions separately. Add the numerators. Simplify. Compared to the estimate, the answer is reasonable.
5 8
_1
+1 8 _
3 6 7 _ or 7 _ 8 4
Find each sum. Write in simplest form. 3 1 4 1 2A. 3_ + 7_ 2B. 6 _ + 9_ 5
5
10
10
Subtract Like Fractions The rule for subtracting fractions with like denominators is similar to the rule for addition. Subtracting Like Fractions To subtract fractions with like denominators, subtract the numerators and write the difference over the denominator.
Words Symbols
a-b a b _ - _ = _, where c ≠ 0 c
c
EXAMPLE
5 5-1 1 4 Example _ - _ = _ or _
c
7
7
7
7
Subtract Fractions
9 13 Find _ -_ . Write the difference in simplest form. 20 20 9 - 13 9 13 _-_=_ 20 20 20 -4 1 = _ or -_ 5 20
1 1 Estimate _ - 1 = -_ 2
2
The denominators are the same. Subtract the numerators. Simplify.
Find each difference. Write in simplest form. 5 10 3 6 3A. _ -_ 3B. _ -_ 15
EXAMPLE Alternative Method You can check your answer by subtracting the whole numbers and fractions separately. 2 9_ - 5_ = (9 - 5) 6 6 2 1 + _ -_ 1
(6 6) 1 = 4 + (-_) 6
= 3_ 5 6
9
15
9
Subtract Mixed Numbers
1 2 Evaluate a - b if a = 9 _ and b = 5 _ . 6
1 2 a - b = 9_ - 5_
6 6 55 32 _ _ = 6 6 23 5 _ = or 3 _ 6 6
6
Estimate 9 - 5 = 4
2 Replace a with 9_ and b with 5_ . 1 6
6
Write the mixed numbers as improper fractions. Subtract the numerators. Simplify.
3 7 4. Evaluate x - y if x = 5 _ and y = 9 _ . 8
8
Personal Tutor at pre-alg.com Lesson 5-5 Adding and Subtracting Like Fractions
251
You can use the same rules for adding or subtracting like algebraic fractions as you did for adding or subtracting like numerical fractions.
EXAMPLE
Add Algebraic Fractions
n 5n Find _ +_ . Write the sum in simplest form. 8 8 n + 5n 5n n _+_=_ 8 8 8 6n 3n = _ or _ 8 4
The denominators are the same. Add the numerators. Add the numerators. Simplify.
Find each sum. Write in simplest form.
y 5y 5B. _ + _
4d 2d 5A. _ +_ 10
Examples 1–3 (pp. 250–251)
8
Find each sum or difference. Write in simplest form. 5 1 1. _ +_
9
10
10
2 4 3. _ + -_
3 3 2. _ +_
7
7
9
3 1 + 8_ 4. 2_
4 2 5. -2_ + -_
3 5 6. 3_ + 6_
3 11 -_ 7. _
5 1 8. -_ -_
4 10 9. _ - _
6
6
5
14
14
Example 4
8
10
8
5
12
8
8
8
12
1 inches tall at the end of school in June. He 10. MEASUREMENT Hoai was 62_ 8
7 inches tall in September. How much did he grow during the was 63_
(p. 251)
8
summer? 3 1 and v = 6_ . ALGEBRA Evaluate each expression if u = 7_ 7
11. u - v Example 5 (p. 252)
ALGEBRA Find each sum or difference. Write in simplest form. 6r 2r 13. _ + _ 11
HOMEWORK
HELP
For See Exercises Examples 15–18 1 19–22 2 23–26 3 27–36 4 37–40 5
7
12. v - u 19 12 _ 14. _ a - a ,a≠0
11
Find each sum or difference. Write in simplest form. 2 1 15. _ +_ 5
5
13 9 18. -_ + -_
21. 24. 27. 30.
16 16 5 7 5_ + 3_ 9 9 5 17 _ -_ 18 18 3 5 2_ - 1_ 8 8 6 5 -8_ - -2_ 11 11
252 Chapter 5 Rational Numbers
3 3 17. -_ + -_
19.
20.
22. 25. 28.
3 7 16. _ +_
31.
10 10 2 2 7_ + 4_ 5 5 5 7 2_ + 2_ 12 12 1 7 _ - -_ 12 12 9 1 8_ - 6_ 10 10 5 3 -4_ -_ 8 8
23. 26. 29. 32.
4 4 9 17 5_ + 5_ 20 20 10 8 _ -_ 11 11 9 7 _ - -_ 20 20 5 4 7_ - 2_ 7 7 3 2 12_ - 13 _ 6 6
8 1 11 ALGEBRA Evaluate each expression if x = _ , y = 2_ , and z = _ . Write in 12 12 12 simplest form.
33. x + y
34. z + y
35. z - x
36. y - x
ALGEBRA Find each sum or difference. Write in simplest form. x 4x 37. _ +_ 8
3r 3r 38. _ +_
8
10
4 1 39. 5_ c - 3_ c 7
10
5 1 40. -2_ y + 8_ y
7
6
6
41. CARPENTRY A 3-foot long shelf is to be installed between two walls that 5 inches apart. How much of the shelf must be cut off so that it fits are 32_ 8
between the walls? Real-World Career Carpenter Carpenters must be able to make precise measurements and know how to add and subtract fractional measures.
For more information, go to pre-alg.com.
PETS The table shows the weight of Leon’s dog during its first five years. 42. How much weight did Leon’s dog lose between
Age Weigh (years) (pounds)
ages 3 and 4? 43. How much weight did Leon’s dog gain between
1
2 17 _
2
5 18 _
8
8 4 8 3 _ 18 8 7 _ 20 8
19 _
3
years 1 and 5? 7 pounds between 44. Suppose Leon’s dog gained 2_
4
8
years 5 and 6. How much does it weigh now?
5
Find each sum or difference. Write in simplest form. 3 5 7 - 7_ + 2_ 45. 12_ 8
EXTRA
PRACTIICE
See pages 771, 798. Self-Check Quiz at pre-alg.com
H.O.T. Problems
8
5 5 1 46. 5_ + 3_ - 2_
8
6
6
6
47. GARDENING Tate’s flower garden has a perimeter of 25 feet. He plans to add 2 feet 9 inches to the width and 3 feet 9 inches to the length. What is the new perimeter in feet? 48. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would add or subtract like fractions. 49. OPEN ENDED Write a subtraction expression in which the difference of two 18 . fractions is _ 25
3 1 and -4_ . Who is 50. FIND THE ERROR Kayla and Ethan are adding -2_ 8 8 correct? Explain your reasoning.
Kayla
-17 35 3 -2 _1 + -4 _ = _ + -_ 8 8 8 8
Ethan
3 35 17 1 -2_ + (-4_) = _ + (- _) 8
52 = -_ or -6 _1 8
2
CHALLENGE The 7-piece square puzzle at the right is called a tangram. 51. If the value of the entire puzzle is 1, what is the value of each piece? 52. How much is A + B? 53. How much is F + D? 54. How much is C + E? 55. Which pieces each equal the sum of E and G?
8
8 8 18 1 _ = - or -2_ 4 8
B C D
A E F
Lesson 5-5 Adding and Subtracting Like Fractions Tony Freeman/PhotoEdit
G
253
56.
Writing in Math Explain how fractions are important when taking measurements. Include in your answer some real-world examples in which fractional measures are used.
57. The average times it takes Miguel to cut his lawn and his neighbor’s lawn are given in the table. Last summer, he cut his lawn 10 times and his neighbor’s 5 times. About how many hours did he spend cutting both lawns? Lawn
_3
Neighbor’s
_2
2
16
15 layer of padding _ inch thick is 16
placed on top. What is the total thickness of the wood and the padding? 3 F 1_ in.
8 1 G 1_ in. 2 24 H 1_ in. 16 1 J 2_ in. 2
Time of Cut (hours)
Miguel’s
1 A 8_ h
9 58. A piece of wood is 1_ inches thick. A
4 4
B 9h
1 C 9_ h
D 10 h
2
Find each quotient. Write in simplest form. (Lessons 5-4) 3 1 59. _ ÷_ 6
4
5 1 60. -_ ÷_ 3
8
2 1 61. _ ÷ 1_ 5
2
4 62. 8_ ÷ -12 5 15
Find each product. Write in simplest form. (Lesson 5-3) 2 _ 63. _ ·3 5
4
8 1 64. _ · -_ 6
9
4 1 65. _ · 2_ 7
1 1 66. -1_ · 1_
3
7
3
67. ALGEBRA Find the product of 4y2 and 8y5. (Lesson 4-5) EXERCISE The table shows the amount of time Craig spends jogging every day. He increases the time he jogs every week. (Lesson 3-7) 68. Write an equation to show the number of minutes spent jogging m for each week w. 69. How many minutes will Craig jog during week 9?
7EEK
PREREQUISITE SKILL Use exponents to write the prime factorization of each number or monomial. (Lesson 4-2) 70. 60 73. 12n 254 Chapter 5 Rational Numbers
71. 175 74.
24s2
72. 112 75. 42a2b
4IME *OGGING MIN
Factors and Multiples Many words used in mathematics are also used in everyday language. You can use the everyday meaning of these words to better understand their mathematical meaning. The table shows both meanings of the words factor and multiple. Term
Everyday Meaning
Mathematical Meaning
factor
something that contributes to the production of result • The weather was not a factor in the decision. • The type of wood is one factor that contributes to the cost of the table.
one of two or more numbers that are multiplied together to form a product
multiple
involving more than one or shared by many • multiple births • multiple ownership
the product of a quantity and a whole number
Source: Merriam Webster’s Collegiate Dictionary
When you count by 2, you are listing the multiples of 2. When you count by 3, you are listing the multiples of 3, and so on, as shown in the table below. Number
Factors
Multiples
2
1, 2
2, 4, 6, 8, . . .
3
1, 3
3, 6, 9, 12, . . .
4
1, 2, 4
4, 8, 12, 16, . . .
Notice that the mathematical meaning of each word is related to the everyday meaning. The word multiple means many, and in mathematics, a number has infinitely many multiples.
Reading to Learn 1. Write your own rule for remembering the difference between factor and multiple. 2. RESEARCH Use the Internet or a dictionary to find the everyday meaning of each word listed below. Compare them to the mathematical meanings of factor and multiple. Note the similarities and differences. a. factotum
b. multicultural
c. multimedia
3. Make lists of other words that have the prefixes fact- or multi-. Determine what the words in each list have in common. Reading Math Factors and Multiples
255
CH
APTER
5
Mid-Chapter Quiz Lessons 5-1 through 5-5
Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1) 4 1. _ 25 1 3. 3_ 8
5 _ 13. _ · 4 18
15 1 _ 15. -1_ ·2 2 3
2 2. -_
9 5 4. 1_ 6
5. MULTIPLE CHOICE Which fraction is between _5 and _7 ? (Lesson 5-1) 7
Find each product or quotient. Write in simplest form. (Lessons 5-3 and 5-4)
8 C _
9 10 D _ 11
10 3 B _ 4
6. MANUFACTURING A garbage bag has a thickness of 0.8 mil, which is equal to 0.0008 inch. What fraction of an inch is this? (Lesson 5-2) Write each decimal as a fraction or mixed number in simplest form. (Lesson 5-2) 7. -6.75
8. 0.12
9. -0.5555 . . .
10. 3.08 3
1 of Earth’s 11. GEOGRAPHY Africa makes up _ 5 entire land surface. Use the table to find the fraction of Earth’s land surface that is made up by each of the other continents. Write each fraction in simplest form. (Lesson 5-2)
Antarctica
Decimal Portion of Earth’s Land 0.095
Asia
0.295
Europe
0.07
North America
0.16
Source: Incredible Comparisons
12. TRAVEL One of the fastest commuter trains is the Japanese Nozomi, which averages 162 miles per hour. About how many minutes would it take to travel 119 miles from Hiroshima to Kokura on the train? (Lesson 5-3) 256 Chapter 5 Rational Numbers
4
1 16. 3 _ ÷ (-4) 3
2
18. MULTIPLE CHOICE If the newsletter is 4 -inch by printed on 8_ 8 11-inch paper and the space between the columns and both ends of the page are shown, how wide are the three equal columns? (Lesson 5-4) 1 F 2_ in.
6 5 G 2_ in. 12
Î ° n
Ó ° n
Ó ° n
Î ° n
1 H 2_ in.
2 3 J 2_ in. 4
Find each sum or difference. (Lesson 5-5) 8 2 19. _ +_
6 11 20. _ -_
1 2 21. -2_ -_ 3 3
5 22. -3 + -4_
3 6 23. 5 _ + 2_
5 7 24. 2_ - 8_
15
7
Continent
8
17. SEWING How many 9-inch ribbons can be cut 1 yards of ribbon? (Lesson 5-4) from 1_
8
7 A _
7 1 14. _ ÷ -_
15
7
12
12
12
8
12
25. MULTIPLE CHOICE A pitcher of lemonade 9 full at the beginning of the party. It was _ 10 _ was only 1 full after the party ended. How 10 much lemonade was drunk during the party? (Lesson 5-5) 1 A _
10 3 B _ 10 3 C _ 5 _ D 4 5
5-6
Least Common Multiple
Main Ideas
Interactive Lab pre-alg.com
• Find the least common multiple of two or more numbers.
A voter voted for both president and a senate seat in the year 2004. a. List the next three years in which the voter can vote for president.
• Find the least common denominator of two or more fractions.
b. List the next three years in which the voter can vote for the same senate seat.
New Vocabulary multiple common multiples least common multiple (LCM) least common denominator (LCD)
Candidate
Length of Term (years)
President
4
Senator
6
c. What will be the next year in which the voter has a chance to vote for both president and the same senate seat?
Least Common Multiple A multiple of a number is a product of that number and a whole number. Sometimes numbers have some of the same multiples. These are called common multiples. multiples of 4:
0, 4, 8, 12, 16, 20, 24, 28, …
multiples of 6:
0, 6, 12, 18, 24, 30, 36, 42, …
Some common multiples of 4 and 6 are 0, 12, and 24.
The least of the nonzero common multiples is called the least common multiple (LCM). So, the LCM of 4 and 6 is 12. When numbers are large, an easier way of finding the least common multiple is to use prime factorization. The LCM is the smallest product that contains the prime factors of each number.
EXAMPLE
Find the LCM
Find the LCM of 108 and 240. Number Prime Factorization
Prime Factors If a prime factor appears in both numbers, use the factor with the greatest exponent.
Exponential Form
108
2·2·3·3·3
22 · 33
240
2·2·2·2·3·5
24 · 3 · 5
The prime factors of both numbers are 2, 3, and 5. Multiply the greatest power of 2, 3, and 5 appearing in either factorization. LCM = 24 · 33 · 5 = 2160
1. Find the LCM of 120 and 180. Extra Examples at pre-alg.com
Lesson 5-6 Least Common Multiple
257
The LCM of two or more monomials is found in the same way as the LCM of two or more numbers.
The LCM of Monomials
EXAMPLE
Find the LCM of 18xy2 and 10y. 18xy2 = 2 · 32 · x · y2 10y = 2 · 5 · y
Find the prime factorization of each monomial. Highlight the greatest power of each prime factor.
LCM = 2 · 32 · 5 · x · y2 = 90xy2
Multiply the greatest power of each prime factor.
2. Find the LCM of 24 a3b and 30a.
Least Common Denominator The least common denominator (LCD) of two or more fractions is the LCM of the denominators.
EXAMPLE
Find the LCD
5 11 Find the LCD of _ and _ .
9 21 Write the prime factorization of 9 and 21.
9 = 32 21 = 3 · 7
Highlight the greatest power of each prime factor.
LCM = 32 · 7 = 63
Multiply.
5 11 The LCD of _ and _ is 63. 9
21
3 7 3. Find the LCD of _ and _ . 8
10
One way to compare fractions is to write them using the LCD. We can multiply the numerator and the denominator of a fraction by the same number, because it is the same as multiplying the fraction by 1.
EXAMPLE Replace
Compare Fractions
1 with , or = to make _
7 _ a true statement.
6
15
The LCD of the fractions is 2 · 3 · 5 or 30. Rewrite the fractions using the LCD and then compare the numerators. 5 1·5 5 _1 = _ =_ Multiply the fraction by _ to make the denominator 30. 5 6 2·3·5 30 7 · 2 14 7 2 _ = _ = _ Multiply the fraction by _ to make the denominator 30. 2 3·5·2 30 15 5 14 1 7 Since _ < _, then _ < _. 30 30 6 15
4. Replace
258 Chapter 5 Rational Numbers
2 with , or = to make _ 3
_5 a true statement. 9
Order Rational Numbers TRAVEL The table shows the arrival times of four flights compared to their scheduled arrival times into Greensboro, North Carolina. Order the flights from least delayed to most delayed. (Hint: A negative fraction indicates a flight that arrived earlier than its scheduled arrival time.)
$EPARTURE #ITY
Step 1 Order the negative fractions first. The LCD of 6 and 8 is 24. 3 9 = -_ -_ 8
1 4 -_ = -_ 6
24
$IFFERENCE BETWEEN !RRIVAL 4IME AND 3CHEDULED 4IME H
!TLANTA
? n
$ALLAS
?
-IAMI
? n
3AN &RANCISCO
?
24
9 3 4 1 < -_, then -_ < -_. Compare the negative fractions. Since -_ Ordering Rational Numbers You can break ordering rational numbers into two steps since negative numbers are always less than positive numbers.
24
8
24
6
Step 2 Order the positive fractions. The LCD of 2 and 7 is 14. 5 10 1_ = 1_ 7
1 7 1_ = 1_ 2
14
14
10 5 7 1 < 1_, then 1_ < 1_. Compare the positive fractions. Since 1_ 14
2
14
7
5 3 1 Since -_ < -1 < 1_ < 1_, the order of the flights from least delayed to most 8
6
2
7
delayed are Atlanta, Miami, San Francisco, and Dallas. Order the fractions from least to greatest. 1 3 1 2 3 3 4 1 5A. -7_ , -6_ , -6_ , -7_ 5B. _, _, _, _ 5 5 4 4 6 20 10 7 Personal Tutor at pre-alg.com
Example 1 (p. 257)
Example 2 (p. 258)
Example 3 (p. 258)
Find the least common multiple (LCM) of each pair of numbers. 1. 6, 8
2. 7, 9
Find the least common multiple (LCM) of each pair of monomials. 4. 36ab, 4b
(p. 258)
Example 5 (p. 259)
5. 5x2, 12y2
6. 14e3, 8e2
Find the least common denominator (LCD) of each pair of fractions. 1 _ 7. _ ,3 2 8
Example 4
3. 10, 14
Replace each 1 10. _ 4
3 _ 16
3 _ 9. _ ,5
2 _ 8. _ , 7 3 10
5 8
with , or = to make a true statement. 10 11. _ 45
_2
5 12. _
9
7
13. WEATHER The table shows the amount of rain that fell during a rainstorm in four Kentucky cities. Order the cities from least to greatest amount of rainfall.
City Bowling Green Frankfort Lexington Louisville
_7 9
Rainfall (in.) 7 1
10 2 15 3 17 3 2 8
Lesson 5-6 Least Common Multiple
259
HOMEWORK
HELP
For See Exercises Examples 14–23, 29 1 24–28 2 30–37 3 38–43 4 44–47 5
Find the least common multiple (LCM) of each pair of numbers or monomials. 14. 4, 10
15. 20, 12
16. 2, 9
17. 16, 3
18. 15, 75
19. 21, 28
20. 14, 28
21. 20, 50
22. 18, 32
23. 24, 32
24. 20c, 12c
25. 16a2, 14ab
26. 7x, 12x
27. 75n2, 25n4
28. 20ef, 52f 3
29. AUTO RACING One driver can circle a one-mile track in 30 seconds. Another driver takes 20 seconds. If they both start at the same time, in how many seconds will they be together again at the starting line? Find the least common denominator (LCD) of each pair of fractions. 1 _ ,7 30. _
4 8 4 _ 34. _ , 5 9 12
Replace each
8 _ 31. _ ,1
4 _ 32. _ ,1
15 3 3 _ 35. _ ,5 8 6
2 _ 33. _ ,6
5 2 1 _ 36. _ ,4 3 7
5 7 5 _ 37. _ ,8 6 9
with , or = to make a true statement.
1 38. _
5 _ 2 12 21 _1 41. _ 100 5
7 39. _
_5 9 6 17 _ 1 42. _ 34 2
3 40. _
_4 5 7 12 _ 36 43. _ 17 51
Order the fractions from least to greatest. 5 _ 1 _ , 3, _ ,5 44. _
12 4 3 6 1 4 11 1 46. -2_ , -2_ , -2_ , -2_ 2 9 6 18
23 4 2 7 45. -_ , -_ , -_ , -_
30 5 3 10 5 _ 1 _ 47. 1_ , 1 3 , 1_ , 11 24 4 8 3
48. PETS In Brady’s math class, approximately 3 of the students have pets. 5 About 41 out of every 50 students in his school have pets. Do a greater fraction of students have pets in Brady’s math class or in his school? 49. ANALYZE TABLES The table shows the number of children who signed up to play soccer in the park district. Would you use the GCF or LCM to find the greatest number of teams that can be formed if each team must have the same number of 6-year-olds, 7-year-olds, and 8-year-olds? Explain your reasoning and then find the answer. How many 6-year-olds, 7-year-olds, and 8-year-olds are on each team?
Age
Number
6
60
7
96
8
24
Find the least common multiple (LCM) of each set of numbers. 50. 7, 21, 84 EXTRA
PRACTIICE
See pages 772, 798. Self-Check Quiz at pre-alg.com
51. 9, 12, 15
52. 45, 30, 35
53. FITNESS Suppose you run on the treadmill every other day and lift weights every third day. After you add pushups to your routine, you do all three exercises every thirtieth day. How often do you do pushups? 54. Find two composite numbers between 10 and 20 whose least common multiple (LCM) is 36.
260 Chapter 5 Rational Numbers
55. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would compare fractions. 56. OPEN ENDED Write two fractions whose least common denominator (LCD) is 35.
H.O.T. Problems
CHALLENGE Determine whether each statement is sometimes, always, or never true. Give an example or explanation to support your answer. 57. The LCM of three numbers is one of the numbers. 58. If two numbers do not contain any factors in common, then the LCM of the two numbers is 1. 59. The LCM of two numbers, except 1, is greater than the GCF of the numbers. 60. The LCM of two whole numbers is a multiple of the GCF of the same two numbers. 61.
Writing in Math Use the information about prime factors on page 257 to explain how to use them to find the LCM of two or more numbers.
62. A radio station is giving away two concert tickets to every sixteenth caller and a dinner for two to every twentieth caller. Which caller will receive both the concert tickets and the dinner?
63. A party goods store sells the party supplies in the table. In order to have the same number of cups, plates, and napkins, what is the least number of each that must be purchased? Quantity in One Package
A 32th
Supply
B 40th
cups
15
C 56th
plates
30
D 80th
napkins
20
F 40
G 48
H 60
J 64
Find each sum or difference. Write in simplest form. (Lesson 5-5) 3 7 64. _ -_ 8
8
9 5 65. 3_ -_ 11
11
13 3 66. _ +_ 14
14
5 1 67. 2_ + 4_ 6
6
ALGEBRA Find each quotient. Write in simplest form. (Lesson 5-4) 3 _1 68. _ n÷n
x x 69. _ ÷_ 8
6
ac 70. _ ÷ _c 5
d
6k 3 71. _ ÷_ 7m
14m
72. ALGEBRA Translate the sum of 7 and two times a number is 11 into an equation. Then find the number. (Lesson 3-6)
PREREQUISITE SKILL Estimate each sum. (page 751) 3 3 73. _ + _ 8
4
9 14 74. _ + _ 10
15
4 1 75. _ + 2_ 7
5
7 2 76. 5 _ +_ 8
3
Lesson 5-6 Least Common Multiple
261
EXTEND
5-6
Algebra Lab
Juniper Green
Juniper Green is a game that was invented by a teacher in England.
GETTING READY This game is for two people, so students should divide into pairs.
RULES OF THE GAME • The first player selects an even number from the hundreds chart and circles it with a colored marker. • The next player selects any remaining number that is a factor or multiple of this number and circles it. • Players continue taking turns circling numbers, as shown below. • When a player cannot select a number or circles a number incorrectly, then the game is over and the other player wins. 2nd move Player 2 circles 7 because it is a factor of 42.
1
1st move Player 1 circles 42.
2
3
4
5
6
7
8
9
10
11 12 13 14
15 16
17 18
19 20
21 22 23 24
25 26
27 28
29 30
31 32 33 34
35 36
37 38
39 40
41 42 43 44
45 46
47 48
49 50
51 52 53 54
55 56
57 58
59 60
61 62 63 64
65 66
67 68
69 70
71 72 73 74
75 76
77 78
79 80
81 82 83 84
85 86
87 88
89 90
91 92 93 94
95 96
97 98
99 100
3rd move Player 1 circles 70 because it is a multiple of 7.
ANALYZE THE RESULTS Play the game several times and then answer the following questions. 1. Why do you think the first player must select an even number? Explain. 2. Describe the kinds of moves that were made just before the game was over. Reprinted with permission from Mathematics Teaching in the Middle School, copyright (c) 1999, by the National Council of Teachers of Mathematics. All rights reserved.
262 Chapter 5 Rational Numbers Geoff Butler
5-7 Main Ideas • Add unlike fractions. • Subtract unlike fractions.
Adding and Subtracting Unlike Fractions 1 1 The sum _ +_ is modeled at the 2 3 right. We can use the LCM to find the sum.
£ Ó
a. What is the LCM of the denominators? b. If you divide the model into six parts, what fraction of the model is shaded?
£ Î
1 _ ? 1? c. How many parts are _ 2 3
d. Describe a model that you could 1 1 use to add _ and _ . Then use it to 3 4 find the sum.
£ Ó
£ Î
Add Unlike Fractions Fractions with different denominators are called unlike fractions. In the activity, you used the LCM of the denominators to rename the fractions. You can use any common denominator. Adding Unlike Fractions To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify as with like fractions.
Words
1 _ 2 1 5 2 3 _ + =_·_+_·_
Example
3
5
3
5
5
3
5 6 11 = _ + _ or _ 15
EXAMPLE LCD You can rename unlike fractions using any common denominator. However, it is usually simpler to use the least common denominator.
15
15
Add Unlike Fractions
Find 1 + 23. 4
_1 + _2 = _1 · _3 + _2 · _4 3
4
3 4 3 3 8 _ _ = + 12 12 11 _ = 12
Find each sum. 1 1 1A. _ +_ 2
5
4
Use 4 · 3 or 12 as the least common denominator. Rename each fraction with the common denominator. Add the numerators.
2 1 1B. _ +_ 3
8
Lesson 5-7 Adding and Subtracting Unlike Fractions
263
EXAMPLE
Add Fractions and Mixed Numbers
Find each sum. Write in simplest form. 3 -7 1 1 + _ Estimate _ - _ = 0 a. _
Negative Signs When adding or subtracting a negative fraction, place the negative sign in the numerator.
2 2 8 12 3 _ -7 -7 _ _3 + _ =_ ·3+ _ · 2 The LCD is 23 · 3 or 24. 2 8 3 12 12 8 9 -14 =_ + _ Rename each fraction with the LCD. 24 24 5 = -_ Add. Compare to the estimate. Is the answer reasonable? 24 1 2 b. 1_ + -2_ Estimate 1 + (-2) = -1 3 9 2 1 11 7 1_ + -2_ =_ + -_ Write the mixed numbers as improper fractions. 9 3 9 3 11 7 _ =_ + -_ · 3 Rename -_73 using the LCD, 9. 3 9 3 11 -21 =_+ _ Simplify. 9 9 -10 1 =_ or -1_ Add. Compared to the estimate, the answer is reasonable. 9 9
3 5 2A. _ +_ 4
9
3 5 2C. 3 _ + -4 _
5 8 2B. -_ +_
14
5
12
6
Subtract Unlike Fractions The rule for subtracting fractions with unlike denominators is similar to the rule for addition. Subtracting Unlike Fractions To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify as with like fractions.
EXAMPLE Reasonableness Use estimation to check whether your answer is reasonable. 6
2 - ≈0-1 21 7
≈ -1 17 is close to -1. - 21
Subtract Fractions and Mixed Numbers
Find each difference. Write in simplest form. 6 1 -_ a. _
7 21 6 _ 1 1 _ _ - 6 =_ -_ ·3 7 7 3 21 21 18 1 =_ -_ 21 21 -17 17 _ = or -_ 21 21
3 8 3A. _ -_ 4
264 Chapter 5 Rational Numbers
9
1 1 b. 6 _ - 4_ The LCD is 21. Rename using LCD.
5
2
5
Write as improper fractions.
2
5
13 _ 21 _ =_ · 5 -_ · 2 Rename using LCD.
2 5 5 65 42 =_-_ 10 10 23 3 = _ or 2 _ 10 10
Subtract.
5 1 3B. 7_ - 6_ 6
2
13 1 1 21 6_ - 4_ =_ -_
8
2
Simplify. Subtract.
5 1 3C. 5 _ - -4 _ 3
9
COMPUTERS To set up a computer network in an office, a 100-foot cable is cut and used to connect 3 computers to the server as shown. How much cable is left to connect the third computer?
You know that the 100-foot cable was used to connect two computers to the server.
Plan
Add the measures of the cables that were already used and subtract that sum from 100. Estimate your answer. 100 - (19 + 41) ≈ 100 - 60 or 40 feet
Solve
3 6 1 1 19_ + 40_ = 19_ + 40_ 4
8 7 = 59_ 8
8
FT 4FSWFSS
Explore
8
FT
3ERVER
Rename 40_ with the LCD, 8. 3 4
Simplify.
8 7 7 100 - 59_ = 99_ - 59_
8 Rename 100 with the LCD, 8, as 99_. 8 8 8 1 = 40_ Simplify. 8 1 There is 40_ feet of cable left to connect the third computer. 8 1 Since 40_ is close to 40, the answer is reasonable. 8
8
Check
3 4. INSECTS The speed of a hornet is 13_ miles per hour. The speed of a 10
4 miles per hour. How much faster is the dragonfly than dragonfly is 17 _
the hornet?
5
Personal Tutor at pre-alg.com
Examples 1, 2 (pp. 263–264)
Find each sum. Write in simplest form. 3 1 1. _ +_
1 1 2. _ +_
5 15 5 1 4. 8_ + 11_ 12 4
Example 3 (p. 264)
3 10 3 5 5. 4_ + 10_ 8 12
1 2 7. _ -_ 3
7 - 2 8. -
10 15 5 1 11. 6_ - 2_ 6 3
3 1 10. -9_ - -5_ 2 4
(p. 265)
6
18
3 4 6. 6_ + -1_ 5
4
Find each difference. Write in simplest form. 4
Example 4
7 1 3. -_ +_
5 7 9. _ -_ 8
12
3 1 12. 12_ - 6_ 2 8
5 1 yards of fabric to make a skirt and 14_ yards to 13. SEWING Jessica needs 5_ 8
make a coat. How much fabric does she need in all?
Extra Examples at pre-alg.com
2
Lesson 5-7 Adding and Subtracting Unlike Fractions
265
HOMEWORK
HELP
For See Exercises Examples 14–19 1, 2 20–23 2 24–31 3 32–33 4
Find each sum or difference. Write in simplest form. 3 3 14. _ +_ 5 4
8
1 4 20. 8_ + 3_ 2 5 3
7 1 21. _ + 4_
1 11 22. -4_ + -7_
5 1 24. _ -_
7 2 25. _ -_
1 2 27. -_ -_
8 2 28. -6_ -_
8
2 7 23. -10_ + 9_
7
21 3 1 19. -_ +_ 7 4
3
8
3 5 5 1 30. 16_ - 12_ 6 3
16
7 29. 216 -_ 30 15
6
24
8
12
3 7 26. _ - -_ 8
5 10 16. _ + -_
13 3 1 18. -_ +_ 8 2
3 5 17. _ + -_
9 3 15. _ +_ 26
10
18
5
3
9
1 1 31. 3_ - -7_ 2 3
For Exercises 32–35, select the appropriate operation. Justify your selection. Then solve. 32. EARTH SCIENCE Did you know that water has a greater density than ice? Use the information in the table to find how much more water weighs per cubic foot.
1 Cubic Foot
Weight (lb)
water
1 2 9 56 10
33. PUBLISHING The length of a page in a yearbook is
ice
1 inch, and the bottom 10 inches. The top margin is _ 2 3 _ margin is inch. What is the length of the page inside
62
4
the margins? 1 of the votes and Sara 34. VOTING In the class election, Murray received _ 3
2 of the votes. Makayla received the rest. What fraction of the received _ 5
votes did Makayla receive? 35. ANALYZE TABLES Use the table to find the sum of precipitation that fell in Columbia, South Carolina, in August, September, and October.
EXTRA
PRACTICE
See pages 772, 798. Self-Check Quiz at pre-alg.com
36. RESEARCH Use the Internet or another source to find out the monthly rainfall totals in your community during the past year. How much rain fell in August, September, and October during the past year?
!MOUNT OF 0RECIPITATION IN
!UG
??
3EPT
??
/CT
?
Find each difference. Write in simplest form.
3 3 - -4_ 37. -19_ 8
H.O.T. Problems
-ONTH
5
5 13 39. 8_ - -12_
2 4 38. -3_ - -2_
4
7
12
40. OPEN ENDED Write a real-world problem that you could solve by 3 1 from 15_ . subtracting 2_ 8
4
3 cup, 41. CHALLENGE A set of measuring cups has measures of 1 cup, _ 4
1 1 1 1 _ cup, _ cup, and _ cup. How could you get _ cup of milk by using 2
3
these measures? 266 Chapter 5 Rational Numbers
4
6
18
9 7 42. FIND THE ERROR Roberto and Daniel are finding _ +_ . Who is correct so 10 12 far? Explain your reasoning.
Daniel 9 + 7 = 9+7 10 12 10 + 12
Roberto
7 7 9 9 12 _ +_=_·_ + _ · 10 12
10
43.
10
12
12
10
Writing in Math Explain how to add and subtract fractions with different denominators. Illustrate your answer with an example using the LCM and an explanation of how prime factorization can be used to add and subtract unlike fractions.
45. The results of a grocery store survey are listed in the table. Find the fraction of families who grill out more than one time per month.
44. For an art project, Halle needs 113 inches of red ribbon and 67 inches 8 9 of white ribbon. Which is the best estimate for the total amount of ribbon that she needs? A 8 in. B 10 in. C 18 in.
2 F _
D 26 in.
25
How Often Do You Grill Out? Times per Fraction of Month People Less than 1
11 50
1
2 25
2–3
4 25
4 or more
27 50
23 G _
7 H _
100
39 J _ 50
10
Find the LCD of each pair of fractions. (Lesson 5-6) 4 _ 46. _ , 7
5 _ 47. _ , 3
9 12
3 _ 48. _ , 2
8 14
1 _ 49. _ , 73
15t 5t
3n 6n
Find each sum or difference. Write in simplest form. (Lesson 5-5) 3 3 50. 2_ + 6_ 4
4
3 2 51. 3_ -_ 5
5 1 52. 4_ + 5_
5
6
6
5 1 53. 6_ - -8_ 4
15
54. MOVIES A movie is made up of hundreds of thousands of individual pictures called frames. The frames are shown through a projector at a rate of 24 frames per second. How many frames would be needed for a 30-minute scene? (Lesson 1-1)
PREREQUISITE SKILL Find each quotient. Round to the nearest tenth, if necessary. (Page 749) 55. 25.6 ÷ 3
56. 37 ÷ 4.7
57. 30.5 ÷ 11.2
58. 46.8 ÷ 15.6
59. 34.8 ÷ 5.8
60. 63 ÷ 7.5
Lesson 5-7 Adding and Subtracting Unlike Fractions
267
5-8
Solving Equations with Rational Numbers
Main Idea • Solve equations containing rational numbers.
Musical sounds are made by vibrations. If n represents the number of vibrations for middle C, then the approximate vibrations for the other notes going up the scale are given below. Notes
Middle C
D
E
F
G
A
B
C
n
9 n 8
5 n 4
4 n 3
3 n 2
5 n 3
15 n 8
2 n 1
Number of Vibrations
a. A guitar string vibrates 440 times per second to produce the A above middle C. Write an equation to find the number of vibrations per second to produce middle C. If you multiply each side by 3, what is the result? b. How would you solve the second equation you wrote in part a? c. How can you combine the steps in parts a and b into one step? d. How many vibrations per second are needed to produce middle C?
Solve Addition and Subtraction Equations You can solve equations with rational numbers using the same properties you used to solve equations with integers.
EXAMPLE Look Back
Solve by Using Addition and Subtraction
a. Solve 2.1 = t - 8.5.
To review solving equations, see Lessons 3-3 and 3-4.
2.1 = t - 8.5
Write the equation.
2.1 + 8.5 = t - 8.5 + 8.5 Add 8.5 to each side. 10.6 = t 3 2 b. Solve x + _ =_ . 5 3 3 2 x+_ =_ 3 5 3 3 3 2 x+_ -_ =_ -_ 5 5 3 5 9 10 1 _ x= -_ or _ 15 15 15
Solve each equation. 1A. n - 9.7 = -13.9 268 Chapter 5 Rational Numbers
Simplify.
Write the equation. Subtract _ from each side. 3 5
Rename the fractions using the LCD and subtract.
7 2 1B. _ +p=_ 10
5
Solve Multiplication and Division Equations Use the same process to solve these equations as those involving integers.
EXAMPLE
Solve by Using Division
Solve -3y = 1.5. Check your solution. -3y = 1.5
Write the equation.
-3y 1.5 _ =_ -3 -3
Divide each side by -3.
y = -0.5
Simplify. Check the solution.
Solve each equation. Check your solution. 2A. 6a = -8.4 2B. -36 = -5z 1 1 To solve _ x = 3, you can divide each side by _ or multiply each side 2
1 by the multiplicative inverse of _ , which is 2.
2
2
Reciprocals
_1 x = 3 2
Recall that dividing by a fraction is the same as multiplying by its multiplicative inverse.
1 2·_ x =2·3
The product of any number and its multiplicative inverse is 1.
EXAMPLE
2
x =6
Write the equation. Multiply each side by 2. Simplify.
Solve by Using Multiplication
-2 Solve _ x = -7. Check your solution. 3
-2 _ x = -7
3 -3 _ -3 -2 _ x =_ (-7) 2 2 3 1 21 x=_ or 10_ 2 2
Write the equation. Multiply each side by -3 . 2 Simplify. Check the solution.
Solve each equation. Check your solution. 5 1 3A. 10 = _ h 3B. _ m = -3 6
8
3 7 PETS Oscar feeds his dog _ cup of dog food in the morning and _ cup 8 4 of dog food in the evening. If a bag of dog food contains 50 cups, how many days will the bag last?
The amount of dog food that Oscar feeds his dog each day is
_7 + _3 = _7 + _6 or 1_5 cups. 8
4
8
8
8
(continued on the next page) Extra Examples at pre-alg.com
Lesson 5-8 Solving Equations with Rational Numbers
269
1_ cups per day 5 8
Words
times
d days
equals
50 cups of dog food
d
=
50
Let d the number of days.
Variable
1_ 5 8
Equation
·
5 1_ d = 50
8 13 _ d = 50 8 8 _ 8 _ · 13 d = 50 · _ 13 8 13 400 d=_ ≈ 30.8 13
Write the equation. Rename 1_ as an improper fraction. 5 8
Multiply each side by _. 8 13
Simplify.
The bag of dog food will last approximately 31 days.
4. RETAIL A pair of shoes that normally costs $45 now costs $30. What fraction of the regular price is the reduced price? Personal Tutor at pre-alg.com
Solve each equation. Check your solution. Example 1 (p. 268)
Examples 2, 3 (p. 269)
1. y + 3.5 = 14.9 4. b - 5 = 13.7
5 5 3 _ _ 5. c - = 6 5
7. -8.4 = -6f
8. 3.5a = 7
Example 4
HOMEWORK
HELP
For See Exercises Examples 14–19 1 20–25 2 26, 27 3 28–31 4 32, 33 5
1 1 3. 4 _ = r + 6_
2
6
1 s = 15 10. -_
4
6. a - 2.7 = 3.2 9. -3.4 = 0.4x
3 11. 9 = _ g 4
6
(pp. 269–270)
3 3 2. _ =w+_
2 12. _ p = -22 3
13. SPACE The weight of an object on the Moon is one-sixth its weight on Earth. If an object weighs 54 pounds on the Moon, how much does it weigh on Earth? Write and solve a multiplication equation to determine the weight of an object on Earth.
Solve each equation. Check your solution. 14. y + 7.2 = 21.9
15. 4.7 = a + 7.1
2 1 16. _ =_ +b
5 7 17. m + _ = -_
3 5 18. 3_ + n = 6_
1 1 19. y + 1_ = 3_
12
18
8
4
3
8
3
18
20. x - 5.3 = 8.1
21. n - 4.72 = 7.52
8 2 23. x - _ = -_
1 1 24. b - 1_ = 4_
3 1 22. n - _ =_ 8 6 1 2 _ 25. 7 = r - 5_
26. 4.1p = 16.4
27. -0.4y = 2
2 28. 8 = _ d
1 29. _ t=9
1 30. 4 = -_ q
3 31. -6 = _ a
5
5
270 Chapter 5 Rational Numbers
15
2
8
4
2
3
5
3
-
Þ
33. BUSINESS A store is going out of business. Items that normally cost $24.99 now cost $16.66. What type of discount is the store offering on those items?
Î Ó £ ä
Õ Ì À i > > Ì Ã À > `
> Þ " Þ « V 9 Ãi , Ìi V Þ Õ Ì > Ã 9i ÜÃ Ì i
.UMBER OF 6ISITORS MILLIONS
For Exercises 32 and 33, write an equation and solve the problem. 32. ANALYZE GRAPHS The graph )''* Dfjk M`j`k\[ shows the most visited national L%J% EXk`feXc GXibj £ä parks in the United States in 2003. The Great Smoky n Mountains had 5.25 million Ç more visitors than the Grand È Canyon. How many people x { visited the Grand Canyon?
53 .ATIONAL 0ARKS
-ÕÀVi\ 4OP 4EN OF %VERYTHING
Solve each equation. Check your solution. 1 2 n=_ 34. _ 3
5 1 35. _ = -_ r
9
8
28 7 36. -_ t = -_
2
9
36
For Exercises 37–40, write an equation and solve the problem. 37. METEOROLOGY When a storm struck, the barometric pressure was 28.79 inches. Meteorologists said that the storm caused a 0.36-inch drop in pressure. What was the pressure before the storm? 38. MUSIC Carla downloaded some songs onto her digital music player and 5 1 full. If the player was _ full before the download, what now the player is _ 6
5
fraction of the space on the player do the new songs occupy? 1 batches of cookies for a bake sale and used 39. COOKING Gabriel made 2_ 2
3 cups of sugar. How much sugar is needed for one batch of cookies? 3_ 4
1 40. PUBLISHING A newspaper is 12_ inches wide and 22 inches long. This is 4
1 inches narrower and one-half inch longer than the old edition. What 1_ 4
were the previous dimensions of the newspaper? EXTRA
PRACTICE
See pages 772, 798. Self-Check Quiz at pre-alg.com
H.O.T. Problems
3 square inches. 41. GEOMETRY The area A of the triangle is 33_ 4
1 bh to find the height h of the Use the formula A = _ 2
triangle with the given base b.
h
42. FIND THE DATA Refer to the United States Data File on 9 in. pages 18–21. Choose some data and write a real-life problem in which you would solve equations with rational numbers. 43. FIND THE ERROR Grace and Ling are solving 0.3x = 4.5. Who is correct? Explain your reasoning. Grace 0.3x = 4.5 0.3x _ _ = 4.5 3 3 x = 1.5
Ling 0.3x = 4.5 0.3x _ _ = 4.5 0.3 0.3 x = 15
Lesson 5-8 Solving Equations with Rational Numbers
271
44. Which One Doesn’t Belong? Identify the equation that does not belong with the other three. Explain your reasoning.
_5 j = 20
c + 3.17 = -3.17
3 2 4_ = w - 2_ 9 4
-0.8k = 10
8
45. CHALLENGE The denominator of a fraction is 4 more than the numerator. If both the numerator and denominator are increased by 1, the resulting 1 . Find the original fraction. fraction equals _ 2
46.
Writing in Math Explain how fractions are used to compare musical notes. How are reciprocals useful in finding the number of vibrations per second needed to produce certain notes? Give an example.
47. GRIDDABLE A hamburger is formed into the shape of a circle with a radius of 3 inches. If a grill is 1_ 4 28 inches wide, how many hamburgers can fit across the grill?
48. Mary and Tabitha ran in a race. Mary’s time was 12 minutes, which 3 of Tabitha’s time. Using t for was _ 4 Tabitha’s time, which equation represents the situation?
Î
£ °
3 A _ t = 12
3 C t-_ = 12
3 B t+_ = 12
3 D 12t = _
4
4
4
4
Find each sum or difference. Write in simplest form. (Lesson 5-7) 3 1 49. _ +_ 5
5 1 50. _ -_
3
6
5 1 + -_ 52. _ 9
1 55. _ 2
1 1 51. -4_ -_
4
3 1 53. -3_ + -2_ 4
12
Replace each
8
6
9 1 54. 8_ - 1_
8
10
6
with , or = to make a true statement. (Lesson 5-6)
5 _ 12
56.
16 _ 9 _ 50 30
4 57. _ 5
48 _ 60
58.
7 _3 _ 8
12
1 59. ALGEBRA Evaluate a - b if a = 9_ and b = 1_ . (Lesson 5-5) 5 6
6
60. GEOMETRY Express the area of the rectangle as a monomial. (Lesson 4-5) 61. HEALTH According to the National Sleep Foundation, teens should get approximately 9 hours of sleep each day. What fraction of the day is this? Write in simplest form. (Lesson 4-4)
PREREQUISITE SKILL Find each sum. (Lesson 2-2) 62. 24 + (-12) + 15
63. (-2) + 5 + (-3)
64. 4 + (-9) + (-9) + 5
65. -10 + (-9) + (-11) + (-8)
272 Chapter 5 Rational Numbers
{X ÓY
XY Ó
EXPLORE
5-9
Algebra Lab
Analyzing Data Often, it is useful to describe or represent a set of data by using a single number. The table shows the daily maximum temperatures for twenty days during a recent September in Phoenix, Arizona. One number to describe this data set might be 96. Some reasons for choosing this number are listed below. • It occurs four times, more often than any other number. • If the numbers are arranged in order from least to greatest, 96 falls in the center of the data set.
There is an equal number of data above and below 96.
80 89 90 94 94 94 95 96 96 96 96 97 97 98 98 98 98 98 99 99 So, if you wanted to describe a typical high temperature for Austin during August, you could say 96°F.
COLLECT THE DATA Collect a group of data. Use one of the suggestions below, or use your own method. • Research data about the weather in your city or in another city, such as temperatures, precipitation, or wind speeds. • Find a graph or table of data in the newspaper or a magazine. Some examples include financial data, population data, and so on. • Conduct a survey to gather some data about your classmates. • Count the number of raisins in a number of small boxes.
ANALYZE THE RESULTS 1. Choose a number that best describes all of the data in the set. 2. Explain what your number means, and explain which method you used to choose your number. 3. Describe how your number might be useful in real life. Explore 5-9 Algebra Lab: Analyzing Data Geoff Butler
273
5-9
Measures of Central Tendency
Main Ideas • Use the mean, median, and mode as measures of central tendency. • Choose an appropriate measure of central tendency and recognize measures of statistics.
New Vocabulary measures of central tendency mean median mode
The Iditarod is a 1150-mile dogsled race across Alaska. The winning times for 1977–2004 are shown.
Winning Times (days)
a. Which number appears most often? b. If you list the data in order from least to greatest, which number is in the middle?
17
15
15
14
12
16
13
13
18
12
11
11
11
11
13
11
11
11
9
9
9
9
10
9
9
8
9
9
Source: Anchorage Daily News
c. What is the sum of all the numbers divided by 28? d. If you had to give one number that best represents the winning times, which would you choose? Explain.
Mean, Median, and Mode When you have a list of numerical data, it is often helpful to use one or more numbers to represent the whole set. These numbers are called measures of central tendency. Measures of Central Tendency mean Mean, Median, Mode • The mean and median do not have to be part of the data set. • If there is a mode, it is always a member of the data set.
the sum of the data divided by the number of items in the data set
median the middle number of the ordered data, or the mean of the middle two numbers mode
the number or numbers that occur most often
a. SPORTS The heights of the players on the girls’ basketball team are shown. Find the mean, median, and mode. sum of heights number of players 63 + 61 + . . . + 59 = __ 12 732 _ or 61 The mean height is 61 inches. = 12
mean = __
Height of Players (in.) 63 58 61 60 61 59 68 55 63 59 66 59
To find the median, order the numbers from least to greatest. 55, 58, 59, 59, 59, 60, 61, 61, 63, 63, 66, 68 60 + 61 _ = 60.5 2
There is an even number of items. Find the mean of the two middle numbers.
The median height is 60.5 inches. The height 59 inches appears three times so 59 is the mode. 274 Chapter 5 Rational Numbers
b. HURRICANES The line plot shows the number of Atlantic hurricanes that occurred each year from 1974 to 2004. Find the mean, median, and mode.
1
⫻
⫻ ⫻ ⫻ ⫻
⫻ ⫻ ⫻ ⫻ ⫻ ⫻
⫻ ⫻ ⫻ ⫻ ⫻
2
3
4
5
⫻ ⫻
⫻ ⫻ ⫻ ⫻
⫻ ⫻ ⫻
⫻ ⫻ ⫻ ⫻
⫻
⫻
6
7
8
9
10
11
12
Source: National Weather Service
2 + 3(4) + 4(6) + 5(5) + 6(2) + 7(4) + 8(3) + 9(4) + 10 + 11 31
mean = _____ ≈ 5.9 Real-World Link A hurricane can be up to 600 miles in diameter and can reach 8 miles in the air. Source: sptimes.com
There are 31 numbers. So the median is the 16th number, or 5. You can see from the graph that 4 occurs most often. So 4 is the mode.
1 , 11, 5, 1. SHOES The shoe sizes of students in Ms. Alberti’s classroom are 10 _ 1 , 6, 6 _ 1 , 11, 7, 7 _ 1 , 8, 9, 5 _ 1 , 10 _ 1 , 4, 10 _ 1 , 10, 5, 14, and 12 _ 1 . Find2the 6, 10 _ 2 2 2 2 2 2 2 mean, median, and mode.
Choose Appropriate Measures Different circumstances determine which of the measures of central tendency are most useful. Using Mean, Median, and Mode mean
• the data set has no extreme values (values that are much greater or much less than the rest of the data)
median • the data set has extreme values • there are no big gaps in the middle of the data mode
• the data set has many repeated numbers
Choose an Appropriate Measure WEATHER The table shows daytime high temperatures for a week. Which measure of central tendency best represents the data? Then find the measure of central tendency. Since the set of data has no extreme values or numbers that are identical, the mean would best represent the data.
Day
Temperature
Sun.
84°F
Mon.
83°F
Tues.
89°F
Wed.
90°F
Thurs.
91°F
84 + 83 + . . . + 80 602 or 86 mean: __ = _
Fri.
85°F
Sat.
80°F
7
7
The temperature 86°F best represents the data.
2. EXERCISE The following set of data shows the number of sit-ups Pablo had done in one minute for the past 6 days: 40, 37, 45, 49, 50, 56. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. Extra Examples at pre-alg.com NASA
Lesson 5-9 Measures of Central Tendency
275
Using measures of central tendency can help you analyze the data from fast-food restaurants. Visit pre-alg.com to continue work on your project.
NUTRITION The table shows the number of Calories per serving of each vegetable. Tell which measure of central tendency best represents the data. Then find the measure of central tendency.
Vegetable
There is one value that is much greater than the rest of the data, 66. Also, there does not appear to be a big gap in the middle of the data. There is only one set of identical numbers. So, the median would best represent the data.
Calories
Vegetable
Calories
asparagus
14
cauliflower
10
beans
30
celery
17
bell pepper
20
corn
66
broccoli
25
lettuce
9
cabbage
17
spinach
9
carrots
28
zucchini
17
9, 9, 10, 14, 17, 17, 17, 20, 25, 28, 30, 66 The median is 17 Calories.
CHECK You can check whether the median best represents the data by finding mean with and without the extreme value. mean with extreme value 262 sum of values __ =_ 12 number of values
Interpreting Data You need to interpret information carefully so that you do not give a false impression for a set of data. As you have seen, extreme values affect how a set of data is perceived.
≈ 21.8
mean without extreme value sum of values 196 __ =_ 11 number of values
≈ 17.8
The mean without the extreme value is closer to the median. The extreme value increases the mean by about 4. Therefore, the median best represents the data.
3. RETAIL An electronics store recorded the number of customers it had each hour during the day. 86, 71, 79, 86, 79, 32, 88, 86, 82, 69, 71, 70 Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency.
Measures of central tendency can be used to show different points of view.
The average wait times for 10 different rides at an amusement park are 65, 21, 17, 52, 25, 17, 11, 22, 60, and 44 minutes. Which measure of data would the park advertise to show the wait times for its rides are short? A Mode B Median
C Mean D Cannot be determined
Read the Test Item To find which measure of central tendency to use, find the mean, median, and mode of the data and select the least measure. 276 Chapter 5 Rational Numbers
Solve the Test Item Analyzing Data Use these clues to help you analyze data. • Extremely high or low values affect the mean. • A value with a high frequency affects the mode. • Data that is clustered affect the median.
65 + 21 + … + 44 10
334 or 33.4 Mean: __ = _
Mode: 17
10
Median: 11, 17, 17, 21, 22, 25, 44, 52, 60, 65 22 + 25 _ or 23.5 2
The mode is the least measure. So the answer is A.
4. Serena received the following scores on her first six math tests: 90, 68, 89, 94, 60, and 93. Which measure of data might she want to use when describing how she is doing in math class? F Mode
G Median
H Mean
J Cannot be determined
Personal Tutor at pre-alg.com
Example 1 (pp. 274–275)
Find the mean, median, and mode for each set of data. If necessary, round to the nearest tenth. 1. 4, 5, 7, 3, 9, 11, 23, 37 ⫻ ⫻
⫻ ⫻ ⫻
⫻ ⫻ ⫻
⫻ ⫻ ⫻ ⫻
1
2
3
4
3.
Example 2 (p. 275)
Example 3 (p. 276)
2. 7.2, 3.6, 9.0, 5.2, 7.2, 6.5, 3.6
5
⫻ ⫻ ⫻
⫻
6
7
8
4. VACATIONS The table shows the number of annual vacation days for nine countries. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 5. BOOKS The number of books sold during the past week is shown below. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 53, 61, 46, 59, 61, 55, 49
Example 4 (pp. 276–277)
Annual Vacation Days Country
Number of Days
Brazil
34
Canada
26
France
37
Germany
35
Italy
42
Japan
25
Korea
25
United Kingdom
28
United States
13
Source: World Tourism Organization
6. MULTIPLE CHOICE Suppose 83 books sold on the eighth day in Exercise 5. Which measure of central tendency would change the most? A The mean B The median C The mode D All measures were affected equally. Lesson 5-9 Measures of Central Tendency
277
HOMEWORK
HELP
For See Exercises Examples 7–12 1 13–14 2, 3 15–17 4
Find the mean, median, and mode for each set of data. Round to the nearest tenth, if necessary. 7. 41, 37, 43, 43, 36
8. 2, 8, 16, 21, 3, 8, 9, 7, 6
9. 14, 6, 8, 10, 9, 5, 7, 13 11. 15
⫻ ⫻ ⫻ 16
10. 7.5, 7.1, 7.4, 7.6, 7.4, 9.0, 7.9, 7.1
⫻
⫻ ⫻ ⫻
⫻ ⫻
⫻ ⫻ ⫻
⫻
17
18
19
20
21
12.
⫻ ⫻ ⫻ ⫻
⫻ ⫻ ⫻ ⫻
⫻ ⫻
⫻
4.1
4.2
4.3
4.4
⫻
22 4.5
4.6
4.7
4.8
For Exercises 13–14, which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 13.
University Michigan
All-Time Football Wins 833
Notre Dame
796
Nebraska
781
Texas
776
Alabama
758
Source: The World Almanac
14.
2003 Corn Production State
Bushels (millions)
CA
27.2
GA
36.8
MD
50.4
MS
71.6
TX
194.7
Source: U.S. Dept. of Agriculture
15. BASKETBALL Refer to the cartoon at the right. Which measure of central tendency would make opponents believe that the height of the team is much taller than it really is? Explain.
EXTRA
PRACTICE
16. TESTS Which measure of central tendency best summarizes the test scores shown below? Explain. 97, 99, 95, 89, 99, 100, 87, 85, 89, 92, 96, 95, 60, 97, 85
See pages 772, 798. Self-Check Quiz at pre-alg.com
H.O.T. Problems
17. ICE SKATING Sasha needs to average 5.8 points from 14 judges to win the competition. The mean score of 13 judges was 5.9. What is the lowest score Sasha can have from the 14th judge and still win? 18. OPEN ENDED Write a set of data with at least four numbers that has a mean of 8 and a median that is not 8. 19. CHALLENGE A real estate guide lists the “average” home prices for counties in your state. Do you think the mean, median, or mode would be the most useful average for homebuyers? Explain. 20.
Writing in Math Explain how measures of central tendency are used in the real world. Include in your answer examples of real-world data from home or school that can be described using the mean, median, or mode.
278 Chapter 5 Rational Numbers
21. The graph shows the number of siblings that Ms. Cantor’s students have. Which measure of data best represents the data?
22. If 18 were added to the data set below, which statement is true? 16, 14, 22, 16, 16, 18, 15, 25 F The mode increases.
ÀÌ iÀÃ >` -ÃÌiÀÃ
G The mean increases.
ÕLiÀ v -ÌÕ`iÌÃ
£ä
H The mode decreases.
n
J The median increases.
È
23. The hourly salaries of employees in a small store are shown. Which measure of data would the store are shown use to attract people to work there?
{ Ó ä
ÀÌV«>Ìi Ì i ,iVÞV} *À}À>¶ Èä ÕLiÀ v -ÌÕ`iÌÃ
HOMEWORK
xÈ
xä {ä Îä £x
Óä £ä
12. HEALTH Seven of the 28 students in math ä class have the flu. Is this sampling of the students who have the flu representative of the entire school? If so, how many of the 464 students who attend the school have the flu?
Real-World Link Approximately 170 million pounds of milk are produced annually in the United States. Source: National Agriculture Statistics Service
EXTRA
PRACTICE
See pages 776, 799. Self-Check Quiz at pre-alg.com
,iëÃi
13. CONCERT As teenagers leave a concert, every 10th person is surveyed. They are asked if they would buy a T-shirt. One hundred forty out of a total of 800 people surveyed said yes. Is this sampling method valid? If so, how many people would you expect to buy T-shirts at the next concert if 7000 attend? Explain your reasoning. 14. ANALYZE TABLES Every hour, twenty customers in a Milk grocery store are randomly selected and surveyed on skim their milk preference. The results are shown in the table. low-fat After reviewing the data, the store manager decided whole that 40% of his total milk stock should be low-fat milk. Is this a valid conclusion? If it is not, what information should the store manager review to make a better conclusion?
Number 88 92 60
15. VIDEOS A video store is considering adding an international movie section. They surveyed 300 random customers, and 80 customers agree the international movie section is a good idea. Should the store add this section? Explain.
346 Chapter 6 Ratio, Proportion, and Percent Lester Lefkowitz/Getty Images
9iÃ
H.O.T. Problems
16. OPEN ENDED Give an example of a biased survey. 17. CHALLENGE Suppose you are a farmer and want to know if your corn crop is ready to be harvested. Describe an unbiased way to determine whether the crop is ready to harvest. 18.
Writing in Math
Why is sampling an important part of the manufacturing process? Illustrate your answer with an unbiased and biased sampling method you can use to check the quality of DVDs.
19. A real estate agent surveys people about their housing preferences at an open house for a luxury townhouse. Which is the best explanation for why the results of this survey might NOT be valid? A The survey is biased because the agent should have conducted the survey by telephone. B The survey is biased because the sample consisted of only people who already are interested in townhouses. C The survey is biased because the sample was a voluntary response sample. D The survey is biased because the agent should have conducted the survey at a single-family home.
20. An online survey produced the following results. If about 38,000 children participated in the survey, about how many drink two cans of soda or less per day? Ü >Þ >à v -`> 9Õ À > >Þ¶ vÛi À Ài £¯ vÕÀ ί Ì Àii ǯ
âiÀ £¯
ÌÜ £x¯
i À iÃà Îǯ
Source: pbskids.org
F 5700
H 14,060
G 8993
J 26,980
21. Find the percent of change from 32 feet to 79 feet. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. (Lesson 6-9) Solve each problem using the percent equation. Round to the nearest tenth. (Lesson 6-8) 22. 7 is what percent of 32? 23. What is 28.5% of 84? ALGEBRA Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 4-4) 17g2h 6r 30x2 12cd 24. _ 25. _ 26. _ 27. _ 15rs 19e 51xy 51g 28. WEATHER During a 10-hour period, the temperature in Browning, Montana, changed at a rate of -10°F per hour, starting at 44°F. What was the ending temperature? (Lesson 2-4) Lesson 6-10 Using Sampling to Predict
347
CH
Study Guide and Review
APTER
6
ownload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
&RACTION $ECIMAL 0ERCENT
Key Concepts Ratios and Rates
(Lesson 6-1)
• A ratio is a comparison of numbers by division. • A unit rate is a simplified rate whose denominator is 1.
Proportions
(Lessons 6-2 and 6-3)
constant of proportionality (p. 298) convenience sample (p. 343) cross product (p. 302) discount (p. 333) interest (p. 334) nonproportional (p. 297) percent (p. 313) percent equation (p. 332) percent of change (p. 338) percent proportion (p. 322)
population (p. 343) proportion (p. 302) proportional (p. 297) ratio (p. 292) rate (p. 293) sample (p. 343) scale (p. 308) scale drawing (p. 308) scale factor (p. 308) scale model (p. 308) unit rate (p. 293)
• A proportion is an equation stating two ratios or c a rates are equal. So, if _ = _ , then ad = bc. b
d
• A proportional relationship exists when the ratios of related terms are equal.
Scale Drawings and Models
(Lesson 6-4)
• A scale drawing or model represents an object that is too large or too small to be drawn or built at actual size. • The ratio of a length on a scale drawing or model to the corresponding length on the real object is the scale factor.
Fractions, Decimals, and Percents
(Lesson 6-5)
• Percent is a ratio of a number to 100. • Fractions, decimals, and percents are all different ways to represent the same number.
Percents
(Lessons 6-6 through 6-9)
part • A percent proportion is _ = percent, where whole
the percent is written as a fraction. • A percent equation is equivalent to a percent proportion except the percent is written as a decimal. • A percent of increase, or decrease, tells how much an amount has increased, or decreased, in relation to the original amount.
348 Chapter 6 Ratio, Proportion, and Percent
Vocabulary Check Complete each sentence with the correct term. Choose from the list above. 1. A statement of equality of two ratios or rates is called a _________. 2. A ________ is a subgroup or subset of the population. 3. A ________ is a ratio of two measurements having different units. 4. The ________ is the amount by which the regular price of an item is reduced. 5. ________ is the amount of money paid or earned for the use of money. 6. Proportional relationships can be described by using the equation y = kx, where k is the ________. 7. A ________ is a type of biased sample. Vocabulary Review at pre-alg.com
Lesson-by-Lesson Review 6–1
Ratios and Rates
(pp. 292–296)
8. 30 hours to 18 hours
Example 1 Express the ratio 2 meters to 35 centimeters as a fraction in simplest form.
9. 10 inches to 4 feet
First, convert 2 meters to centimeters.
Express each ratio or rate as a fraction in simplest form.
200 cm 2m _ =_
10. 5 quarts to 5 gallons
35 cm
11. 2 tons to 1800 pounds
200 cm ÷ 5 40 cm 40 _ =_ or _
12. BASEBALL Jean got 12 hits out of 16 times at bat. Express this rate as a fraction in simplest form.
6–2
35 cm ÷ 5
Proportional and Nonproportional Relationships Determine whether the set of numbers in each table forms a proportion. 13. Boxes 1 2 3 4 Pens
8
14. Number of People Brownies Eaten
16
24
32
2
4
6
8
2
5
7
10
15. FESTIVALS A customer at the ring toss booth gets 8 rings for $2. Write an equation relating the cost to the number of rings. At this same rate, how much would a customer pay for 11 rings? for 20 rings?
6–3
Using Proportions
3 12 9 22.5 18. _ =_ 7 y
7 cm
7
(pp. 297–300)
Example 2 Determine whether the set of numbers in the table forms a proportion. Distance (meters)
30
56
69
80
Time (minutes)
1
2
3
4
Write the rate of distance to time for each minute in simplest form. 30 _
56 _ _ = 28
1
2
1
69 _ _ = 23 3
1
80 _ _ = 20 4
1
Since the rates are not equal, the set of numbers do not form a proportion.
(pp. 302–306)
3 15 =_ . Example 3 Solve _
Solve each proportion. n 4 16. _ =_
35 cm
Next, divide the numerator and denominator by the GCF, 5.
84 21 17. _ =_ x
120 5 0.6 19. _ =_ 7.5 k
20. REAL ESTATE A homeowner whose house is assessed for $120,000 pays $1800 in taxes. At the same rate, what is the tax on a house assessed at $135,500?
15 _3 = _ 7
x
x 7 Write the proportion.
3 · x = 7 · 15 Cross products 3x = 105
Multiply.
3x 105 _ =_
Divide each side by 3.
3
3
x = 35
The solution is 35.
Chapter 6 Study Guide and Review
349
CH
A PT ER
6 6–4
Study Guide and Review
Scale Drawings and Models
(pp. 308–312)
On the model of a ship, the scale is 1 inch = 12 feet. Find the actual length of each room. 21. 22. 23.
Room Stateroom Galley Gym
Model Length 0.9 in. 3.8 in. 6.0 in.
0.25 in. 1.75 in. drawing length drawing length _ =_ actual length 1 ft actual length x ft
0.25 · x = 1 · 1.75
24. MAPS The length of the expressway is 900 miles. If 0.5 inch on a map represents 50 miles, what is the length of the expressway on the map?
6–5
Fractions, Decimals, and Percents
26. 8.8%
27. 120% 28. 87.5%
Express each decimal or fraction as a percent. Round to the nearest tenth percent if necessary. 29. 0.24
30. 1.9
2 31. _ 5
Cross multiply.
0.25x = 1.75
Simplify.
x=7
Divide.
The actual length of the pond is 7 feet.
(pp. 313–318)
Express each percent as a fraction or mixed number in simplest form and as a decimal. 25. 35%
Example 4 A scale drawing shows a pond that is 1.75 inches long. The scale on the drawing is 0.25 inch = 1 foot. What is the length of the actual pond?
6 32. _ 80
Example 5 Express 60% as a fraction in simplest form and as a decimal. 60 3 or _ 60% = _
060% = 0.60 or 0.6
5
100
Example 6 Express 0.38 as a percent. 0.38 = 0.38 or 38% 5 Example 7 Express _ as a percent.
_5 = 0.625 or 62.5%
8
8
33. PETS In a survey, 0.2 of American households own a dog, one-fourth own cats, and 7% own a bird. Which group is largest? Explain.
6–6
Using the Percent Proportion
(pp. 322–326)
Use the percent proportion to solve each problem. 34. 18 is what percent of 45? 35. What is 74% of 110? 36. 23 is 92% of what number? 37. MUSIC Thirty percent of the music that Meghan owns is classical. If Meghan owns 120 albums, how many are classical? 350 Chapter 6 Ratio, Proportion, and Percent
Example 8 Forty-eight is 32% of what number? 48 32 _ =_ b
100
Write the percent proportion.
48 · 100 = b · 32 Find the cross products. 4800 = 32b 150 = b
Simplify. Divide each side by 32.
So, 48 is 32% of 150.
Mixed Problem Solving
For mixed problem-solving practice, see page 799.
6–7
Finding Percents Mentally
(pp. 327–331)
Find the percent of each number mentally. 38. 50% of 86
39. 20% of 55
1 40. 33_ % of 24
41. 90% of 60
3
Example 9 Find 20% of $45 mentally. 1 of 45 20% of 45 = _ 5
Think: 20% = _. 1 5
Think: 15 of 45 is 9.
=9 So, 20% of $45 is $9.
Estimate. Explain which method you used to estimate.
Example 10 Estimate 32% of 150.
42. 48% of 32
43. 67% of 30
1 1 32% is about 33_ % or _ .
1 44. _ % of 304
45. 147% of 200
_1 of 150 is 50.
3
3
3
3
46. BASKETBALL Tito has 244 free throw attempts in his high school career. If he was successful 77% of the time, about how many free throws did he make?
So, 32% of 150 is about 50.
47. GEOGRAPHY The United States has 88,633 miles of shoreline. Of the total amount, 35% is located in Alaska. About how many miles of shoreline are located in Alaska?
6–8
Using Percent Equations
(pp. 332–336)
Solve each problem using the percent equation.
Example 11 119 is 85% of what number?
48. 24 is what percent 50?
The part is 119, and the percent is 85%. Let n represent the whole.
49. What is 12.5% of 68?
119 = 0.85n
Write 85% as the decimal 0.85.
50. 56 is 28% of what number?
119 0.85n _ =_ 0.85 0.85
Divide each side by 0.85.
51. 35.7 is what percent of 17?
140 = n
So, 119 is 85% of 140.
52. SHOPPING A jersey is on sale for 50% off the original price. A week later, the manager takes another 50% off the sale price. Is the jersey now free? Explain. 53. INVESTMENTS What is the interest on 1 years? $10,000 invested at 9% for 1_ 2 Round to the nearest cent.
Chapter 6 Study Guide and Review
351
CH
A PT ER
6 6–9
Study Guide and Review
Percent of Change
(pp. 338–342)
Find the percent of change. Round to the nearest tenth, if necessary. Then state whether each percent of change is a percent of increase or a percent of decrease. 54. from 40 ft to 12 ft
net weight - amount of change = ___ original weight
56. from 80 lb to 77 lb 57. from 29 min to 54 min
≈ -0.611 or -61.1%
58. CLUBS The number of members in the recycling club increased by 15 people. If the club had 12 members previously, what was the percent of increase of the members in the club?
Using Sampling to Predict
percent of change
14 - 36 =_ 36 -22 _ = 36
55. from 80 cm to 96 cm
6–10
Example 12 Find the percent of change from 36 pounds to 14 pounds.
The percent of decrease is about 61.1%.
(pp. 343–347)
Identify each sample as biased or unbiased and describe its type. Explain your reasoning. 59. To determine the weekly top ten songs, the local radio station asks people to log onto their Web site and vote for their favorite song. 60. To determine what type of dessert people in a community like, Sara surveys 20% of the people who enter three different chocolate shops. 61. MUSIC Sixty-three out of the 105 students in the band said that their favorite class was music. Is this sampling representative of the entire school? If so, how many of the 848 students who attend the school would say music is their favorite class?
Example 13 AUTOMOBILES From a batch of 100,000 cars, the manufacturer tests the exhaust on every 500th car. The manufacturer found that 1 car was below standards. Is this sampling method valid? If so, find how many of the 100,000 cars you can expect to be below standards. Explain your reasoning. This is a systematic random sample because the samples are selected according to a specific interval. So, this sampling method is reasonable and will produce a valid prediction. Since every 500 cars were sampled, there were a total of 100,000 ÷ 500 or 200 cars sampled and 1 was substandard. The number of cars that were below standards 1 or 0.5%. were _ 200
Find 0.5% of 100,000. n = 0.005 × 100,000 or 500 Multiply. So, there are approximately 500 substandard cars in the batch. 352 Chapter 6 Ratio, Proportion, and Percent
CH
A PT ER
6
Practice Test
Express each rate as a unit rate. Round to the nearest tenth. 1. 145 miles in 3 hours 2. 245 miles every 6 hours 8.4 1.2 =_ a 3. What value of y makes _ y 1.1 proportion?
20. GIFTS Gifts, Inc. is selling their bobbleheads for 25% off their regular price. If a bobblehead costs $49.95, for how much is it on sale? Find the percent of change. Round to the nearest tenth, if necessary.
4. WATER Which bottle of water costs more per ounce: $1.25 for 12 ounces or $1.50 for 16 ounces?
21. 175 pounds to 140 pounds 22. 1 hour to 1 hour 10 minutes
5. TAXI The taxi cab company charges a $2 fee plus $1.10 for each mile driven. Complete the table and determine whether the pattern forms a proportion.
23. MULTIPLE CHOICE A builder is designing a swimming pool that is 8.5 inches in length on the scale drawing. The scale of the drawing is 1 inch = 6 feet. What is the length of the actual swimming pool?
Miles Driven Cab Fare
1
Express each percent as a fraction or mixed number in simplest form and as a decimal. 6. 36%
7. 225%
8. 0.6%
Express each decimal or fraction as a percent. Round to the nearest tenth, if necessary. 9. 0.47
30 11. _
10. 0.025
A 44 ft
C 49 ft
B 47 ft
D 51 ft
24. TOASTERS To determine the quality of toasters coming off an assembly line, the manager pulls every fiftieth toaster off the line and toasts a piece of bread. Identify this sample as biased or unbiased and describe its type. Explain your reasoning.
22
Use the percent proportion to solve each problem.
25. MULTIPLE CHOICE The table lists the reasons shoppers use online customer service.
12. 36 is what percent of 80? 13. 35.28 is 63% of what number? 14. Find 35% of 200. Estimate. Explain which method you used to estimate. 15. 25% of 82
16. 62% of 77
1 % of 2453 17. _
18. 439% of 61
12
19. INVESTMENTS Find the interest on $2700 that 1 years. is invested at 4% for 2_ 2
Chapter Test at pre-alg.com
Reasons
Percent
Track Delivery Product Information Verify Shipping Charges Transaction Help
54 24 17 5
Out of 350 shoppers who own a computer, how many would you expect to say they use online customer service to track packages? F 189
H 84
G 154
J 19
Chapter 6 Practice Test
353
CH
A PT ER
Standardized Test Practice
6
Cumulative, Chapters 1–6
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. Of the coins in a piggy bank, 20% are quarters, 35% are dimes, 15% are nickels, and 30% are pennies. There are 200 coins in the bank altogether. Which proportion can be used to find q, the total number of quarters in the piggy bank? 20 200 A_ =_ q 100 q 20 =_ B _ 200 100 20 _ = 200 C _ q
3. GRIDDABLE Ana earns $6.80 per hour when she works on weekdays. She earns twice that amount per hour when she works on weekends. If Ana worked 4 hours on Tuesday, 4 hours on Thursday, and 5 hours on Saturday, then how much did she earn in dollars? 4. The table shows the number of new car sales at a car dealership over the past several months. Month
4
5
6
7
8
9
10
11
Number of Sales
28
35
40
37
33
31
29
41
Suppose the dealership has a special promotion and sells 84 cars during December. Which measure of data will change the most?
100 q 20 _ =_ D 300 100
2. David surveyed 50 golfers on a Saturday morning about their favorite outdoor activity. The results are shown in the table below.
A The mean B The median C The mode D All measures will be affected equally.
Favorite Outdoor Activity Activity Hiking Golf Swimming Other
Number of Votes 8 20 9
Question 4 If you are unsure of the correct answer, eliminate the choices you know are incorrect. Then consider the remaining choices.
13
Based on these results, David concluded that playing golf is the favorite outdoor activity among people in his city. Which is the best explanation for why his conclusion might not be valid? F The survey should have been done on different days of the week. G The survey should have been done with men and women golfers. H The sample was not representative of all the people in the city. J There are more golfers on the weekend than during the week. 354 Chapter 6 Ratio, Proportion, and Percent
5. A statistician is organizing the winning percentages of the top hockey teams in the league. Choose the group of percentages that is listed in order from greatest to least. F 0.518, 0.517, 0.524, 0.508 G 0.524, 0.518, 0.517, 0.508 H 0.508, 0.524, 0.518, 0.517 J 0.508, 0.517, 0.518, 0.524 6. GRIDDABLE Nina wants to buy a new pair of inline skates. The regular price of the skates is $90, but they are on sale this week for 15% off. What is the sale price of the skates in dollars? Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
7. A commemorative coin was worth $4.50 when it was issued in 2000. The table shows the value of the coin several years after it was issued.
10. Four-fifteenths of a flower bouquet had 6 of the bouquet had yellow flowers and _ 15 red flowers. What part of the bouquet was NOT yellow or red? 4 F _
Value of Commemorative Coin Year
Value of Coin
2000
$4.50
2001
$5.00
2002
$5.75
2003
$6.75
2004
$8.00
2005
$9.50
3 H_
5 _ G 11 15
5
1 J _ 3
11. In 2003, a new planet was discovered beyond Pluto. This new planet is 1010 miles from the Sun. Which of the following represents this number in standard notation? A 100 mi B 10,000 mi
Based on the information in the table, what is a reasonable prediction for the value of the coin in 2009? A $15.50
C $18.75
B $18.00
D $19.25
C 10,000,000 mi D 10,000,000,000 mi
Pre-AP Record your answers on a sheet of paper. Show your work.
8. The Milky Way galaxy is made up of about 200 billion stars, including the Sun. Write this number in scientific notation. F 2.0 × 108
H 2.0 × 1010
G 2.0 × 109
J 2.0 × 1011
12. An electronics store is having a sale on certain models of televisions. Mr. Castillo would like to buy a television that is on sale. This television normally costs $679.
9. The total number of points scored by 12 players on a basketball team is 870 points for the season. Craig scored 184 points. Which equation can be used to find p, the average number of points scored by Craig’s teammates?
a. What price, not including tax, will Mr. Castillo pay if he buys the television on Saturday?
870 + 184 A p=_ 11 870 - 184 B p=_
b. What price, not including tax, will Mr. Castillo pay if he buys the television on Wednesday?
11 870 - 184 C p=_ 11 184 D p = 870 + _ 11
c. How much money will Mr. Castillo save if he buys the television on Saturday?
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
Go to Lesson...
6-2
6-10
1-1
5-9
5-2
6-8
1-7
4-8
5-9
5-5
4-8
6-8
Chapter 6 Standardized Test Practice
355
Functions and Graphing
7 •
Identify proportional or nonproportional linear relationships in problem situations and solve problems.
•
Make connections among various representations of a numerical relationship.
•
Use graphs, tables, and algebraic representations to make predictions and solve problems.
Key Vocabulary direct variation (p. 378) function (p. 359) rate of change (p. 371) slope (p. 384)
Real-World Link INSECTS If x is the number of chirps a cricket makes every 15 seconds, the equation y = x + 40 can help you estimate y, the outside temperature in degrees Fahrenheit.
Functions and Graphing Make this Foldable to collect examples of functions and graphs. Begin with an 11’’ × 17’’ sheet of paper.
1 Fold the short sides so they meet in the middle.
3 Open and Cut along the second fold to make four tabs. Staple a sheet of grid paper inside.
356 Chapter 7 Functions and Graphing Cisca Casteljins/Foto Natura/Minden Pictures
2 Fold the top to the bottom.
4 Add axes as shown. Label the quadrants on the tabs.
X
/
s
W
GET READY for Chapter 7 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Express each relation as a table. Then determine the domain and range. (Used in Lesson 7-1)
1. {(0, 4), (-3, 3)}
2. {(-5, 11), (2, 1)}
3. {(6, 8), (7, 10), (8, 12)}
Example 1
Express the relation {(1, 2), (2, 1), (3, 12)} as a table. Then determine the domain and range. The domain is {(1, 2, 3)}. x y
4. MARATHON Kelly ran at a rate of 8 feet per second. What is the distance Kelly ran in 24 seconds? in 1 minute 15 seconds?
The range is {(2, 1, 12)}.
1
2
2
1
3
12
Use the coordinate plane to name the point for each ordered pair. (Used in Lesson 7-5) y 5. (-3, 0) B 6. (3, -2) A 7. (-4, -2) 8. (0, 4) C F 9. (4, 6) x O 10. (4, 0) D E
Example 2
11. DIRECTIONS On a coordinate plane, the movie theater is located at (5, -6) and the grocery store is located at (-2, 6). Write directions on how to walk from the movie theater to the grocery store.
Point N is at (-3, 2).
Write an equation that describes each sequence. (Used in Lesson 7-8) 12. 4, 5, 6, 7, . . . 13. 13, 14, 15, 16, . . .
Example 3
Use the coordinate plane to name the point for (-3, 2). y Step 1 Start at (0, 0). K
Step 2 Step 3
Move 3 units to the left.
M
N
x
O
Move 2 units up.
L J
14. FITNESS Becky started an exercise program that calls for 12 minutes of jogging each day during the first week. Each week thereafter, Becky increases the time she jogs by 5 minutes. In which week will she first jog more than 30 minutes?
Write an equation that describes the sequence 2, 5, 8, 11, 14, . . . Term Number (n)
1
2
3
4
Term (t)
2
5
8
11
The difference of the term number is 1. The difference in the terms is 3. So, common difference is 3 times the difference in the term numbers minus 1. So, t = 3n - 1. Chapter 7 Get Ready for Chapter 7
357
EXPLORE
7-1
Algebra Lab
Input and Output In a function, there is a relationship between two quantities or sets of numbers. You start with an input value, apply a function rule of one or more operations, and get an output value. In this way, each input is assigned exactly one output.
ACTIVITY Step 1 To make a function machine, draw three squares in the middle ⫺5 of a 3-by-5-inch index ⫺4 Input Rule Output card, shown here in ⫺3 blue. ⫺5 ⫻2⫹3 ⫺7 ⫺2 Step 2 Cut out the square on ⫺1 the left and the square 0 on the right. Label the 1 left “window” INPUT 2 and the right 0 3 “window” OUTPUT. 1 4 Step 3 Write a rule such as 2 “× 2 + 3” in the center 3 square. 4 Step 4 On another index card, list the integers from -5 to 4 in a column close to the left edge. Step 5 Place the function machine over the number column so that -5 is in the left window. Step 6 Apply the rule to the input number. The output is -5 × 2 + 3, or -7. Write -7 in the right window.
ANALYZE THE RESULTS 1. Slide the function machine down so that the input is -4. Find the output and write the number in the right window. Continue this process for the remaining inputs. 2. Suppose x represents the input and y represents the output. Write an algebraic equation that represents what the function machine does. 3. Explain how you could find the input if you are given a rule and the corresponding output. 4. Determine whether the following statement is true or false. Explain. The input values depend on the output values. 5. Write an equation that describes the relationship between the input value x and output value y in each table.
Input -1 0 1 3
Output ⫺2 0 2 6
Input ⫺2 ⫺1 0 1
6. Write your own rule and use it to make a table of inputs and outputs. Exchange your table of values with another student. Use the table to determine each other’s rule. 358 Chapter 7 Functions and Graphing
Output 2 3 4 5
7-1
Functions
Main Ideas • Determine whether relations are functions. • Use functions to describe relationships between two quantities.
New Vocabulary function vertical line test
The table shows the time it should take a scuba diver to ascend to the surface from several depths to prevent decompression sickness. a. On grid paper, graph the depths and times as ordered pairs (depth, time). b. Describe the relationship between the two sets of numbers.
Depth (ft)
Time (s)
7.5 15 22.5 30
15 30 45 60
Source: diverssupport.com
c. If a scuba diver is at 45 feet, what is the best estimate for the amount of time she should take to ascend? Explain.
Relations and Functions Recall that a relation is a set of ordered pairs. A function is a special relation in which each member of the domain is paired with exactly one member in the range. Not a Function {(-2, 1), (-2, 3), (-5, 4), (-9, 7)}
Function {(-2, 1), (-4, 3), (-5, 4), (-9, 7)} Ó { x
£ Î { Ç
Ó
£ Î { Ç
x
Review Vocabulary
This is a function because each domain value is paired with exactly one range value
domain the set of x-coordinates in a relation; Example: The domain of {(1, 4), (-3, -7)} is {1, -3}.
Since functions are relations, they can be represented using ordered pairs, tables, or graphs.
range the set of y-coordinates in a relation; Example: The range of {(1, 4), (-3, -7)} is {4, -7}. (Lesson 1-6)
EXAMPLE
This is not a function because -2 in the domain is paired with two range values, 1 and 3.
Ordered Pairs and Tables as Functions
Determine whether each relation is a function. Explain. a. {(-3, 1), (-2, 4), (-1, 7), (0, 10), (1, 13)} This relation is a function because each element of the domain is paired with exactly one element of the range. b.
x y
5 1
3 3
2 1
0 3
-4 -2
1A. {(5, 1), (6, 3), (7, 5), (8, 0)}
-6 2
This is a function because for each element of the domain, there is only one corresponding element in the range. 1B.
x y
-1
-6
-3
-1
-5
-2
7
6
2
8
-2
1
Personal Tutor at pre-alg.com Lesson 7-1 Functions
359
Vocabulary Link Function Everyday Use a relationship in which one quality or trait depends on another. Height is a function of age. Math Use a relationship in which a range value depends on a domain value, y is a function of x.
Another way to determine whether a relation is a function is to apply the vertical line test to the graph of the relation. Use a pencil or straightedge to represent a vertical line.
y
Place the pencil at the left of the graph. Move it to the right across the graph. If, for each value of x in the domain, it passes through no more than one point on the graph, then the graph represents a function.
EXAMPLE
x
O
Use a Graph to Identify Functions
Determine whether the graph at the right is a function. Explain your answer.
y
The graph represents a relation that is not a function because it does not pass the vertical line test. At least one input value has more than one output value. By examining the graph, you can see that when x = 2, there are three different y values.
O
x
2. Determine whether the graph of times and distances below is a function. Explain your answer.
Time (min) 4 12 16 20 28
Distance (m) 1 3 4 5 7
Distance (mi)
Describe Relationships A function describes the relationship between two quantities such as time and distance. For example, the distance you travel on a bike depends on how long you ride the bike. In other words, distance is a function of time. 9 8 7 6 5 4 3 2 1 0
y
x 4 8 12 16 20 2428 32 36 Time (min)
SCUBA DIVING The table shows the water pressure as a scuba diver descends.
Depth (ft)
a. Do these data represent a function? Explain. Real-World Link Most of the ocean’s marine life and coral live and grow within 30 feet of the surface. Source: scuba.about.com
This relation is a function because at each depth, there is only one measure of pressure. b. Describe how water pressure is related to depth. Water pressure depends on the depth. As the depth increases, the pressure increases.
360 Chapter 7 Functions and Graphing Peter/Stef Lamberti/Getty Images
0 1 2 3 4 5
Water Pressure (lb/ft2) 0 62.4 124.8 187.2 249.6 312.0
Source: infoplease.com
Extra Examples at pre-alg.com
3. SALES Do these data in the table represent a function? Describe how price is related to the number of balloons purchased. Number of Balloons Price per Balloon
100 $0.99
200 $0.90
300 $0.79
400 $0.60
500 $0.50
Determine whether each relation is a function. Explain. Example 1 (p. 359)
1. {(13, 5), (-4, 12), (6, 0), (13, 10)}
2. {(9.2, 7), (9.4, 11), (9.5, 9.5), (9.8, 8)}
3. Domain Range
4.
3 -2 5 -4 3
-3 -1 0 1 2
Example 2
5.
x 5 2 -7 2 5
y 4 8 9 12 14
6.
y
y
(p. 360)
O
Example 3 (pp. 360–361)
HOMEWORK
HELP
For See Exercises Examples 9–16 1 17–20 2 21–24 3
x
x
O
MEASUREMENTS For Exercises 7 and 8, use the data in the table. 7. Do the data represent a function? Explain. 8. Is there any relation between foot length and height?
Name Remana
Foot Length (cm) 24
Height (cm) 163
Enrico
25
163
Jahad
24
168
Cory
26
172
Determine whether each relation is a function. Explain. 9. {(-1, 6), (4, 2), (2, 36), (1, 6)}
10. {(-2, 3), (4, 7), (24, -6), (5, 4)}
11. {(9, 18), (0, 36), (6, 21), (6, 22)}
12. {(5, -4), (-2, 3), (5, -1), (2, 3)}
13. Domain
14.
-4 -2 0 3
15.
x -7 0 11 11 0
y 2 4 6 8 10
Range -2 1 2 1
16.
Domain -1 -2 -2 -6 x 14 15 16 17 18
Range 5 5 1 1
y 5 10 15 20 25 Lesson 7-1 Functions
361
Determine whether each relation is a function. Explain. 17.
x
O
19.
The lowest wind chill temperature ever recorded at an NFL game was -59°F in Cincinnati, Ohio, on January 10, 1982. Source: Southern AER
EXTRA
PRACTICE
See pages 776, 800. Self-Check Quiz at pre-alg.com
H.O.T. Problems
y
x
FARMING For Exercises 21–24, use the table that shows the number and size of farms in the United States every decade from 1950 to 2000. 21. Is the relation (year, number of farms) a function? Explain. 22. Describe how the number of farms is related to the year. 23. Is the relation (number of farms, average size of farms) a function? Explain. 24. Describe how the average size of farms is related to the year.
x
O
20.
y
O
Real-World Link
18.
y
y
x
O
Farms in the United States Number
Average Size
(millions)
(acres)
1950
5.6
213
1960
4.0
297
1970
2.9
374
1980
2.4
426
1990
2.1
460
2000
2.2
434
Year
Source: U.S. Dept. of Agriculture
Tell whether each statement is always, sometimes, or never true. Explain. 25. A function is a relation. 26. A relation is a function. ANALYZE TABLES For Exercises 27 and 28, use the table that shows how various wind speeds affect the actual temperature of 15°F. 27. Do the data represent a function? Explain. 28. Describe how wind chill temperatures are related to wind speed.
Wind Speed Wind Chill (mph) Temperature (°F) 0 15 10 3 20 -2 30 -5 40 -8
29. RESEARCH Use the Internet or another source to Source: National Weather Service find the complete wind chill table. Does the data for actual temperature and wind chill temperature for a specific wind speed represent a function? Explain.
30. OPEN ENDED Draw the graph of a relation that is not a function. Explain why it is not a function. 31. REASONING Describe three ways to represent a function. Show an example of each. Then describe three ways to represent a relation that is not a function and show an example of each.
362 Chapter 7 Functions and Graphing Craig Tuttle/CORBIS
CHALLENGE The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. 32. Determine whether the inverse of the relation {(4, 0), (5, 1), (6, 2), (6, 3)} is a function. 33. Is the inverse of a function always, sometimes, or never a function? Give an example to explain your reasoning. 34.
Writing in Math
How can the relationship between water depth and time to ascend to the water’s surface be a function? Include a discussion about whether water depth can ever have two corresponding times to ascend to the water’s surface.
35. Which statement is true about the data in the table? A The data represent a function. B The data do not represent a function. C As the value of x increases, the value of y increases.
x -4 2 5 10 12
y -4 16 8 -4 15
D A graph of the data would not pass the vertical line test.
36. The table shows the water temperatures at various depths in a lake. Describe how temperature is related to the depth. Depth (ft) Temperature (°F )
0 74
10 72
20 71
30 61
40 55
50 53
F The water temperature stays the same as the depth increases. G The water temperature decreases as the depth increases. H The water temperature increases as the depth increases. J The water temperature decreases as the depth decreases.
37. TECHNOLOGY The table shows the results of a survey in which middle school students were asked whether they ever used the Internet for the activities listed. Use the data to predict how many students in a middle school of 650 have used the Internet to do research for school. (Lesson 6-10)
Activity E-mail research for school instant message games
Percent 71% 70% 61% 71%
Source: Atlantic Research and Consulting
Find each percent of change. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. (Lesson 6-9) 38. from $56 to $49
39. from 110 mg to 165 mg
40. SCIENCE The length of a DNA strand is 0.0000007 meter. Write the length of a DNA strand using scientific notation. (Lesson 4-7)
Evaluate each expression if x = 4 and y = -1. (Lesson 1-3) 41. 3x + 1 42. 2y 43. y + 6
44. 20 - 4x Lesson 7-1 Functions
363
Graphing Calculator Lab
EXTEND
7-1
Function Tables
You can use a TI-83/84 Plus graphing calculator to create function tables. By entering a function and the domain values, you can find the corresponding range values.
ACTIVITY Use a function table to find the range of y = 3n + 1 if the domain is {-5, -2, 0, 0.5, 4}. Step 1 Enter the function.
Step 2
• The graphing calculator uses X for the domain values and Y for the range values. So, Y = 3X + 1 represents y = 3n + 1.
• Use TBLSET to select Ask for the independent variable and Auto for the dependent variable. Then you can enter any value for the domain.
• Enter Y = 3X + 1 in the Y= list. KEYSTROKES:
3 X,T,,n
Format the table.
KEYSTROKES:
1
2nd [TBLSET]
ENTER
ENTER
Step 3 Find the range by entering the domain values. • Access the table. KEYSTROKES:
2nd [TABLE]
y 3(5) 1 14
• Enter the domain values. KEYSTROKES: -5
ENTER -2 ENTER . . . 4 ENTER
The range is {-14, -5, 1, 2.5, 13}.
EXERCISES Use the [TABLE] option on a graphing calculator to complete each exercise. 1. Suppose you are using the formula d = rt to find the distance d a car travels for the times t in hours given by {0, 1, 3.5, 10}. a. If the rate is 60 miles per hour, what function should be entered in the Y= list? b. Make a function table for the given domain. c. Between which two times in the domain does the car travel 150 miles? d. Describe how a function table can be used to estimate the time it takes to drive 150 miles. 2. Serena is buying one packet of pencils for $1.50 and a number of fancy folders x for $0.40 each. The total cost y is given by y = 1.50 + 0.40x. a. Use a function table to find the total cost if Serena buys 1, 2, 3, 4, and 12 folders. b. Suppose plain folders cost $0.25 each. Enter y = 1.50 + 0.25x in the Y = list as Y2. How much does Serena save if she buys pencils and 12 plain folders rather than pencils and 12 fancy folders? 364 Chapter 7 Functions and Graphing
7-2
Representing Linear Functions
Main Ideas
Interactive Lab pre-alg.com
• Solve linear equations with two variables.
Peaches cost $1.50 per can.
• Graph linear equations using ordered pairs.
a. Complete the table to find the cost of 2, 3, and 4 cans of peaches.
New Vocabulary linear equation
Number of Cans (x )
1.50x
Cost (y )
1
1.50(1)
1.50
2
3 b. On grid paper, graph the ordered pairs 4 (number, cost). Then draw a line through the points.
c. Write an equation representing the relationship between number of cans x and cost y.
Solutions of Equations An equation such as y = 1.50x is called a linear equation. A linear equation in two variables is an equation in which the variables appear in separate terms and neither variable contains an exponent other than 1. Reading Math Input and Output The variable for the input is called the independent variable because the values are chosen and do not depend upon the other variable. The variable for the output is called the dependent variable because it depends on the input value.
Solutions of a linear equation are ordered pairs that make the equation true. One way to find solutions is to make a table. Consider y = -x + 8. y = -x + 8 x Step 1 Choose any convenient values for x.
⎧ -1 0 ⎨ 1 2 ⎩
y = -x + 8
y
(x, y)
y = -(-1) + 8
9
(-1, 9)
y = -(0) + 8
8
(0, 8)
y = -(1) + 8
7
(1, 7)
y = -(2) + 8
6
(2, 6)
Step 2 Substitute the values for x.
⎫ ⎬ ⎭
Step 4 Write the solutions as ordered pairs.
Step 3 Simplify to find the y-values.
So, four solutions of y = -x + 8 are (-1, 9), (0, 8), (1, 7), and (2, 6).
EXAMPLE
Use a Table of Ordered Pairs
Find four solutions of y = 2x - 1. Choose four values for x. Then substitute each value into the equation and solve for y. Four solutions are (0, -1), (1, 1), (2, 3), and (3, 5).
x
y = 2x - 1
y
(x, y)
y = 2(0) - 1
1
(0, -1)
1
y = 2(1) - 1
1
(1, 1)
2
y = 2(2) - 1
3
(2, 3)
3
y = 2(3) - 1
5
(3, 5)
1. Find four solutions of y = x + 5. Lesson 7-2 Representing Linear Functions
365
Solve an Equation for y CELL PHONES Games cost $8 to download onto a cell phone. Ring tones cost $1. Find four solutions of 8x + y = 20 in terms of the numbers of games x and ring tones y Darcy can buy with $20. Explain each solution. First, rewrite the equation by solving for y. 8x + y = 20 Write the equation. 8x + y - 8x = 20 - 8x Subtract 8x from each side. y = 20 - 8x Simplify. Choose four x values and substitute them into y = 20 - 8x. Choosing x-Values
(1, 12)
It is often convenient to choose 0 as an x value to find a value for y.
(2, 4)
→ She can buy 1 game and
12 ring tones. → She can buy 2 games and
4 ring tones.
(_14 , 18)
x
y = 20 - 8x
y
(x, y)
1
y = 20 - 8(1)
12
(1, 12)
2
y = 20 - 8(2)
4
(2, 4)
_1 4
1 y = 20 - 8 _
18
(_14 , 18)
5
y = 20 - 8(5)
-20
(5, -20)
(4)
→ This solution does not make sense in the situation because there
cannot be a fractional number of games.
(5, -20) → This solution does not make sense in the situation because there cannot be a negative number of ring tones.
2. SHOPPING Fancy goldfish x cost $3, and regular goldfish y cost $1. Find three solutions of 3x + y = 8 in terms of the number of each type of fish Tyler can buy for $8. Describe what each solution means. Personal Tutor at pre-alg.com
Graph Linear Equations A linear equation can also be represented by a graph. Linear Equations
Reading Math Linear Equations Graphs of all “linear” equations are straight lines. The coordinates of all points on a line are solutions of the equation.
y
y
y y 1x 3
x
O yx1
x
O
x
O
y 2x
Nonlinear Equations y
y
y
y 2x 3
y x2 1 O
x
O
x
x
O y 3 x
366 Chapter 7 Functions and Graphing
EXAMPLE
Graph a Linear Equation
Graph y = x + 1 by plotting ordered pairs. First, find ordered pair solutions. Four solutions are (-1, 0), (0, 1), (1, 2), and (2, 3).
Plotting Points It is best to find at least three points. You can also graph just two points to draw the line and then graph one point to check.
x
y=x+1
y
(x, y)
-1
y = -1 + 1
(-1, 0)
y=0+1
1
(0, 1)
1
y=1+1
2
(1, 2)
2
y=2+1
3
(2, 3)
Plot these ordered pairs and draw a line through them. Note that the ordered pair for any point on this line is a solution of y = x + 1. The line is a complete graph of the function.
y
(1, 0)
x
O
CHECK It appears from the graph that (-2, -1) is also a solution. Check this by substitution. y=x+1 -1 -2 + 1 -1 = -1
(2, 3) (1, 2) (0, 1)
yx1
Write the equation. Replace x with -2 and y with -1. Simplify.
3. Graph y = 2x - 1 by plotting ordered pairs. A linear equation is one of many ways to represent a function. Representing Functions Words Table of Ordered Pairs
Equation
Example 1 (p. 365)
(p. 366)
Example 3 (p. 367)
x
y
-3
1
-2
2
-1
3
Graph
y
x
O
y x3
y=x-3
Find four solutions of each equation. Show each solution in a table of ordered pairs. 1. y = x + 8
Example 2
The value of y is 3 less than the corresponding value of x.
2. y = 4x
3. y = 2x - 7
4. -5x + y = 6
5. SCIENCE The distance in miles d that light travels in t seconds is given by the linear function d = 186,000t. Find two solutions of this equation and describe what they mean. Graph each equation by plotting ordered pairs. 6. y = x + 3
Extra Examples at pre-alg.com
7. y = 2x - 1
8. x + y = 5
Lesson 7-2 Representing Linear Functions
367
HOMEWORK
HELP
For See Exercises Examples 9–22 1 23–26 2 27–34 3
Copy and complete each table. Use the results to write four solutions of the given equation. Show each solution in a table of ordered pairs. 9. y = x - 9 x
x-9
-1
-1 - 9
10. y = 2x + 6 y
x
2x + 6
-4
2(-4) + 6
4
2
7
4
y
Find four solutions of each equation. Show each solution in a table of ordered pairs. 11. y = x + 4
12. y = x - 7
13. y = 3x
14. y = -5x
15. y = 2x - 3
16. y = 3x + 1
17. x + y = 9
18. x + y = -6
19. 4x + y = 2
20. 3x - y = 10
21. 2x - y = -4
22. -5x + y = 12
MEASUREMENT The equation y = 0.62x describes the approximate number of miles y in x kilometers. 23. Describe what the solution (8, 4.96) means. 24. About how many miles is a 10-kilometer race? FITNESS During a workout, a target heart rate y in beats per minute is represented by y = 0.7(220 - x), where x is a person’s age. 25. Compare target heart rates of people 20 years old and 50 years old. 26. In which quadrant(s) would the graph of y = 0.7(220 - x) make sense? Explain your reasoning. Graph each equation by plotting ordered pairs. 27. y = x + 2
28. y = x + 5
29. y = x - 4
30. y = -x - 6
31. y = -2x + 2
32. y = 3x - 4
33. x + y = 1
34. x - y = 6
ANALYZE TABLES Determine whether each relation or equation is linear. Justify your answer. 35.
Real-World Link Walking is the top sports activity among Americans over the age of 7. Source: Statistical Abstract of the United States
x
y
-1
36.
1
-1
-1
-1
1
1
1
-1
4
2
-1
-2
-1
1
2
2
4
2
39. y = x2
40. y = 5
GEOMETRY For Exercises 41–44, use the following information. The formula for the perimeter of a square with sides s units long is P = 4s. 41. Find three ordered pairs that satisfy this condition. 42. Draw the graph that contains these points. 43. Why do negative values of s make no sense in the context of the situation? 44. Does this equation represent a function? Explain.
368 Chapter 7 Functions and Graphing Duomo/CORBIS
y
y
38. 3x + y = 20
37.
x
x
s s
s s
EXTRA
PRACTICE
See pages 777, 800. Self-Check Quiz at pre-alg.com
H.O.T. Problems
45. MUSIC Aisha has $50 to spend on music. Single songs cost $1 to download and entire CDs cost $10. Find an equation to represent the number of single songs x and the number of CDs y Aisha can buy with $50. Then, find three ordered pairs that satisfy this condition. 46. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would graph a linear equation. 47. CHALLENGE Compare and contrast the functions shown in the tables. (Hint: Compare the change in values for each column.) 48. OPEN ENDED Write and graph a linear equation that has (-2, 4) as a solution.
x
y
x
y
-1
-2
-1
1
1
2
1
1
2
4
2
4
49. NUMBER SENSE Explain why a linear function has infinitely many solutions. 50.
Writing in Math How can linear equations represent a function? Include in your answer a description of four ways that you can represent a function, and an example of a linear equation that could be used to determine the cost of x pounds of bananas that are $0.49 per pound.
Miles Driven d
51. The graph describes the distance d Brock can drive his car on g gallons of gasoline. How many gallons of gas will he need to drive 280 miles? A 14 gal 80 B 16 gal 60 C 17 gal 40 D 19 gal
52. Monica has $440 to pay a painter to paint her bedroom. The painter charges $55 per hour. The equation y = 440 - 55x represents the amount of money left after x number of hours worked by the painter. What does the solution (7, 55) represent? F Monica has $7 left after 55 hours of painting.
20
1
2 3 Gallons g
4
G Monica has $55 left after 7 hours of painting. H The job is completed after 7 hours. J The job is completed after 55 hours.
Determine whether each relation is a function. Explain. (Lesson 7-1) 53. {(0, 6), (-3, 9), (4, 9), (-2, 1)} 54. (-0.1, 5), (0, 10), (-0.1, -5) 55. VOLUNTEERING In a survey of high school students, 28% said they volunteered at least 2 hours a week. In a class of 32 high school students, how many would you predict volunteer at least 2 hours a week? (Lesson 6-10) 56. SHOPPING Two cans of soup cost $0.88. One can costs $0.20 more than the other. How much would 5 cans of each type of soup cost? (Lesson 1-1)
PREREQUISITE SKILL Evaluate each expression. (Lesson 1-2) 18 - 10 57. _ 8-4
16 - 7 58. _ 1-3
46 - 22 59. _ 2005 - 2001
31 - 25 60. _ 46 - 21
Lesson 7-2 Representing Linear Functions
369
Language of Functions Equations that are functions can be written in a form called function notation as shown below. equation
function notation
y = 4x + 10
f(x) = 4x + 10 Read f(x) as f of x.
So, f (x) is simply another name for y. Letters other than f are also used for names of functions. For example, g(x) = 2x and h(x) = -x + 6 are also written in function notation. domain
range
In a function, x represents the domain values, and f(x) represents the range values.
f(x)
=
4 x + 10
f(3) represents the element in the range that corresponds to the element 3 in the domain. To find f(3), substitute 3 for x in the function and simplify.
Read f(3) as f of 3.
f (x) = 4x + 10 f (3) = 4(3) + 10 f (3) = 12 + 10 or 22
Write the function. Replace x with 3. Simplify.
So, the function value of f for x = 3 is 22.
Reading to Learn 1. RESEARCH Use the Internet or a dictionary to find the everyday meaning of the word function. Write a sentence describing how the everyday meaning relates to the mathematical meaning. 2. Write your own rule for remembering how the domain and the range are represented using function notation. 3. Copy and complete the table below. x
f(x) = 3x + 5
f(0) = 3(0) + 5
f(x)
1 2 3
4. If f(x) = 4x - 1, find each value. a. f(2)
b. f(-3)
c. f
_21
5. Find the value of x if f(x) = -2x + 5 and the value of f(x) is -7. 370 Chapter 7 Functions and Graphing
7-3
Rate of Change
Main Ideas • Solve problems involving rates of change.
New Vocabulary rate of change
The graph shows the changes in height and distance of a small airplane during 30 minutes of flight.
À«>i *ÃÌ Ónää
a. Between which two consecutive points did the vertical position of the airplane increase the most? decrease the most? How do you know?
6iÀÌV> *ÃÌ vÌ®
• Find rates of change.
Ó{ää Óäää
#
%
! "
£Èää
&
$
£Óää nää {ää ä
b. What is happening to the airplane between points A and B?
Óää Èää £äää £{ää £nää ÀâÌ> *ÃÌ vÌ®
c. Find the ratio of the vertical change to the horizontal change for each section of the graph. Which section is the steepest?
Rate of Change A rate of change is a rate that describes how one quantity changes in relation to another quantity. The rate of change of the vertical position of the airplane to the horizontal position from point B to point C is shown below. change in vertical position 2400 - 2000 ___ =_ 800 - 400
change in horizontal position
400 or 1 ft in vertical position for every 1 ft =_ 400
in horizontal position
TECHNOLOGY The table shows the growth of subscribers to satellite radio. Find the rate of change from 2003 to 2005. 7.7 - 1.8 rate of change = _ 2005 - 2003
= 2.95
← change in subscribers ← change in time Simplify.
Year 2003 2004 2005
Total Subscribers (millions) 1.8 4.5 7.7
Source: www.govtech.net
So, the rate of change from 2003 to 2005 was an increase of about 2.95 million people per year.
1. TECHNOLOGY Find the rate of change from 2004 to 2005 in the table above. Personal Tutor at pre-alg.com Lesson 7-3 Rate of Change
371
Rates of change can be positive or negative. This corresponds to an increase or decrease in the y-value between the two data points. When a quantity does not change over time, it is said to have a zero rate of change.
EXAMPLE
Compare Rates of Change
GEOMETRY The table shows how the perimeters of an equilateral triangle and a square change as side lengths increase. Compare the rates of change.
Perimeter y Triangle Square 0 0 6 8 12 16
Side Length x 0 2 4
change in y change in x 6 or 3 For each side length increase of 2, =_ the perimeter increases by 6. 2 change in y square rate of change = _ change in x For each side length increase of 2, _ = 8 or 4 the perimeter increases by 8. 2
triangle rate of change = _
y
28 24 20 16 12 8 4
Perimeter (in.)
The perimeter of square increases at a faster rate than the perimeter of a triangle. A steeper line on the graph indicates a greater rate of change for the square.
square
triangle
1 2 3 4 5 6 7 Side Length (in.)
x
2. GEOMETRY The perimeter of a regular hexagon changes as its side lengths increase by 1 inch. Compare this rate of change with the rates of change for the triangle and the square described above.
Real-World Link The temperature of the air is about 3˚F cooler for every 1000 feet increase in altitude. Source: hot-air-balloons.com
Negative Rate of Change EARTH SCIENCE The data points on the graph show the relationship between altitude and temperature. Find the rate of change.
/i«iÀ>ÌÕÀi LÛi -i> iÛi Y
change in temperature = __
Broken Lines In Example 3, there are no data points between the points that represent temperature. So, a broken line was used to help you easily see trends in the data.
change in altitude 4.2°C - 24°C Temperature goes from = __ 3 km - 0 km 24°C to 4.2°C. Altitude goes from 0 km to 3 km. -19.8°C =_ Simplify. 3 km = -6.6°C/km Express as a unit rate.
Ón Ó{ Óä
£È £Ó n
{ ä
£
X
Ó Î { x È Ç ÌÌÕ`i ®
So, the rate of change is -6.6°C/km, or a decrease of 6.6°C per 1-kilometer increase in altitude.
372 Chapter 7 Functions and Graphing David Keaton/CORBIS
/i«iÀ>ÌÕÀi  ®
rate of change
3. Bracy received $200 cash for her birthday. The table shows the amount y remaining after x weeks. Find the rate of change. Interpret its meaning.
Weeks
Amount ($)
x
y
2
$160
4
$120
6
$80
Rates of Change Rate of Change
positive
zero
negative
Real-Life Meaning
increase
no change
decrease
Y
Y
Y
SLANTS UPWARD
SLANTS DOWNWARD HORIZONTAL LINE
Graph X
"
1.
35 30 25 20 15 10 5 0
Example 2 (p. 372)
"
X
Find the rate of change for each linear function. Amount of Water (gal)
(pp. 371–373)
2.
y
1 2 3 4 5 6 7 8 Time (min)
x
Time (h)
Wage ($)
x
y
1
12
2
24
3
36
3. AGE The graph shows the median ages that men got married in different years. Compare the rates of change of the median age between 1999 and 2001 and between 2001 and 2003. iÌÌ} >ÀÀi` ÓÇ°Ó ÓÇ°£ }i Þi>Àî
Examples 1, 3
X
"
ÓÇ°ä ÓÈ° ÓÈ°n ÓÈ°Ç ÓÈ°È ä £Ç £ Óää£ ÓääÎ 9i>À
3OURCE "UREAU OF THE #ENSUS
Lesson 7-3 Rate of Change
373
For See Exercises Examples 4–7 1, 3 8–11 2
Find the rate of change for each linear function. 4.
36 24 12
6.
5.
y
1 2 3 Number of Feet
Time (min)
Temperature (˚C)
HELP
Number of Inches
HOMEWORK
x
Temperature (°F)
y
28 24 20 16 12 8 4 0
7.
x
400 1200 2000 Altitude (m)
Time (h)
Distance (mi) y
x
y
x
58
0.0
1
56
0.5
25
2
54
1.5
75
3
52
3.0
150
TELEVISION The graph shows the percent of households that had cable television in the United States. 8. Find the rate of change in the percent of households from 2001 to 2003. 9. Find the rate of change in the percent of households from 1999 to 2001.
>Li /6 È°{
*iÀViÌ
È°ä
Èn°È Èn°Ó
ÈÇ°n ÈÇ°{
£ Óää£ 9i>À
ÓääÎ
ä £Ç
3OURCE .IELSEN -EDIA 2ESEARCH
10. ANALYZE TABLES The table shows late fees for DVDs and video games at a video store. Compare the rates of change.
EXTRA PRACTICE See pages 777, 800. Self-Check Quiz at pre-alg.com
374 Chapter 7 Functions and Graphing
Days Late x
DVDs
Video Games
2
$3
$4
4
$6
5
$7.50
$8 $10
>vÀ> `ÀÃ £xä
£Óx ÕLiÀ
11. ANALYZE GRAPHS The graph shows the populations of California condors in the wild and in captivity. Write several sentences that describe how the populations have changed since 1965. Include the rate of change for several key intervals.
Late Fee y
£ää Çx
7ILD
xä Óx
#APTIVITY
ä ½Èx ½Çä ½Çx ½nä ½nx ½ä ½x ½ää ½äx 9i>À
3OURCE 53 &ISH AND 7ILDLIFE 3ERVICE
H.O.T. Problems
12. CHALLENGE Describe the rate of change for a graph that is a horizontal line and a graph that is a vertical line. 13. OPEN ENDED Describe a real-world relationship between two quantities that involves a positive rate of change. 14.
Writing in Math Use the data about airplane flight on page 371 to explain how rate of change affects the graph of the airplane’s position.
15. The graph shows the population of Kendall. In which decade was there the greatest population change? £ä]äää ]äää
16. The table shows a relationship between time and altitude of a hot-air balloon. Which is the best estimate for the rate of change for the balloon from 1 to 5 seconds?
*«Õ>Ì
n]äää
Time (s) 1 2 3 4 5
Ç]äää È]äää x]äää {]äää ä
£Èä
£Çä
£nä 9i>À
£ä
Altitude (ft) 6.3 14.5 22.7 30.9 39.1
Óäää
A 1960–1970
C 1980–1990
B 1970–1980
D 1990–2000
F 7.6 ft/s
H 8.2 ft/s
G 7.8 ft/s
J
8.8 ft/s
Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2) 17. y = 2x + 5
18. y = -3x
19. x + y = 7
Determine whether each relation is a function. (Lesson 7-1) 20. {(2, 12), (4, -5), (-3, -4), (11, 0)}
21. {(-4.2, 17), (-4.3, 16), (-4.3, 15), (-4.3, 14)}
Solve each problem by using the percent equation. (Lesson 6-8) 22. 10 is what percent of 50?
23. Find 95% of 256.
24. RAIN A raindrop falls from the sky at about 17 miles per hour. How many feet per second is this? Round to the nearest foot per second. (Lesson 6-1) Find each product. Write in simplest form. (Lesson 5-3) 8 25. 7 _
( 21 )
10 14 26. -_ · -_ 15
28
14 2 27. _ · 3_ 15
7
5 1 28. -1_ · 3_ 4
9
PREREQUISITE SKILL Rewrite y = kx by replacing k with each given value. (Lesson 1-3) 29. k = 5
30. k = -2
31. k = 0.25
1 32. k = _ 3
Lesson 7-3 Rate of Change
375
7-4
Constant Rate of Change and Direct Variation
Main Ideas The graph shows the relationship between time and distance of a car traveling 55 miles per hour.
Travel Time
a. Choose any two points on the graph and find the rate of change.
• Solve problems involving direct variation.
b. Repeat Part a with a different pair of points. What is the rate of change?
New Vocabulary linear relationship constant rate of change direct variation constant of variation
Distance (mi)
• Identify proportional and nonproportional relationships by finding a constant rate of change.
350 300 250 200 150 100 50 0
y
(4, 220) (3, 165) (2, 110) (1, 55) 1 2 3 4 5 6 Time (h)
x
c. MAKE A CONJECTURE What is the rate of change between any two points on the line.
Constant Rates of Change The graph above is a straight line. Relationships that have straight-line graphs are called linear relationships. Notice in the graph above that as the time in hours increases by 1, the distance in miles increases by 55. +1
+1
Time (h)
1
Distance (mi)
55
+1
2
+1
3
4
110 165 220
Rate of Change change in distance 55 __ = _ or 55 miles per hour 1 change in time
+ 55 + 55 + 55 + 55
The rate of change between any two data points in a linear relationship is the same or constant. Another way of describing this is to say that a linear relationship has a constant rate of change. Constant Rate of Change
376 Chapter 7 Functions and Graphing
Not a Constant Rate of Change
Y
Y
"
Constant Rate of Change
X
"
Y
X
"
X
EXAMPLE
Use a Graph to Find a Constant Rate of Change
TECHNOLOGY An Internet advertisement contains a circular icon that decreases in size until it disappears. Find the constant rate of change for the radius in the graph shown. Describe what the rate means.
- À} ÀVi Y ,>`ÕÃ v ÀVi V®
Choose any two points on the line and find the rate of change between them. We will use the points at (2, 5) and (6, 4). (2, 5) → 2 seconds, radius 5 centimeters (6, 4) → 6 seconds, radius 4 centimeters change in radius rate of change = __ change in time 4 cm - 5 cm = __ 6s-2s 1 cm = -_ 4s
= -0.25 cm/s
Ç È x { Î Ó £ £
ä
← centimeters ← seconds
Ó Î { x È Ç X ÕLiÀ v -iV`Ã
The radius goes from 5 cm to 4 cm. The time goes from 2 s to 6 s. Simplify. Express this as a unit rate.
The rate of change -0.25 cm/s means that the radius of the circle is decreasing at a rate of 0.25 centimeter per second.
*iÀiÌiÀ v -µÕ>Ài V®
1. TECHNOLOGY Another icon on the advertisement described above is a square that increases in size. Find the constant rate of change for the perimeter of the square in the graph shown. Describe what the rate means.
ÀÜ} -µÕ>Ài Y x { Î Ó £ ä
£ Ó Î { x X ÕLiÀ v -iV`Ã
Some linear relationships are also proportional. That is, the ratio of each non-zero y-value compared to the corresponding x-value is the same. Look Back To review proportional relationships, see Lesson 6-2.
Number of People x
1
2
3
4
Cost of Parking y
4
8
12
16
cost of parking y 8 16 4 12 __ →_ =4 _ =4 _ =4 _ =4 2 3 1 4 number of people x
The ratios are equal, so the linear relationship is proportional. Number of People x
1
2
3
4
Cost of Tickets y
13
22
31
40
cost of tickets y 31 40 13 22 1 _ __ →_ = 13 _ = 11 _ = 10_ = 10 2 3 3 4 1 number of people x
The ratios are not equal, so the linear relationship is nonproportional. Extra Examples at pre-alg.com
Lesson 7-4 Constant Rate of Change and Direct Variation
377
EXAMPLE
Use Graphs to Identify Proportional Linear Relationships
POOLS The height of the water as a pool is being filled is recorded in the table. Determine if there is a proportional linear relationship between the height of the water and the time.
} 1« > * i} Ì °®
£ä
To determine if the quantities are proportional, height y find _ for points on the graph.
Real-World Link The Johnson Space Center in Houston, Texas, has a 6.2 million gallon pool used to train astronauts for space flight. It is 202 feet long, 102 feet wide, and 40 feet deep. Source: www.jsc.nasa.gov
time x 10 _5 = 2.5 _ = 2.5 2 4
n È { Ó ä
15 20 _ = 2.5 _ = 2.5 6 8
x £ä £x Óä Óx /i °®
Since the ratios are the same, the height of the water is proportional to the time.
2. PLANTS Determine whether the relationship between time and distance of a car on page 376 is a proportional relationship. Personal Tutor at pre-alg.com
Direct Variation A special type of linear equation that describes constant rate of change is called a direct variation. The graph of a direct variation always passes through the origin and represents a proportional linear relationship. Direct Variation Words
Directly Proportional Since k is a constant rate of change in a direct variation, we can say the following. • y varies directly with x. • y is directly proportional to x.
A direct variation is a relationship in which the ratio of y to x is a constant, k. We say y varies directly with x.
Symbols
y = kx, where k ≠ 0
Example
y = 2x
Model
y y 2x
x
O
In the equation y = kx, k is called the constant of variation or constant of y proportionality. The direct variation y = kx can be written as k = _ x . In this form, you can see that the ratio of y to x is the same for any corresponding values of y and x. In other words, x and y vary in such a way that they have a constant ratio, k.
Use Direct Variation to Solve Problems TECHNOLOGY The time it takes to burn amounts of information on a CD is given in the table. a. Write an equation that relates the amount of information and the time it takes. Step 1 Find the value of k using the equation y = kx. Choose any point in the table. Then solve for k. y = kx
Direct variation
10 = k(2.5) Replace y with 10 and x with 2.5. 4=k
378 Chapter 7 Functions and Graphing NASA/JSC
Divide each side by 2.5.
Amount of Information (megabytes)
Time (s)
Rate of changes (MB/s)
x
y
y k=_ x
2.5
10
4
15
4
10
3.75
40
4
25
100
4
Step 2 Use k to write an equation. y = kx Direct variation y = 4x Replace k with 4.
b. Predict how long it will take to fill a 700 megabyte CD with information. y = 4x
Write the direct variation equation.
y = 4(700) Replace x with 700. y = 2800
Multiply.
It will take 2800 seconds or about 46 minutes and 40 seconds to fill a 700 megabyte CD with information.
««iÃ
UNIT COST The graph shows the cost of apples. 3A. Write an equation that relates cost and weight. 3B. Predict how much 5 pounds of apples would cost.
ÃÌ
fÎ
Y Ó] f£°Èä
fÓ f£
ä°x] fä°{ä
fä
£°Óx] f£
X
£ Ó 7i} Ì L®
Î
Proportional Linear Relationships Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change.
Words
B
Graph
a _ is constant and
Symbols
b
change in b _ is constant. change in a
"
Find the constant rate of change for each linear function and interpret its meaning. 1.
ÀÜÌ v *>Ì Ç È x { Î Ó £ ä
2.
-i} vvii
Y
Y Ç /Ì> ->ià f®
(p. 377)
i} Ì v *>Ì °®
Example 1
A
£ Ó Î { x È Ç ÕLiÀ v 7iiÃ
X
È x { Î Ó £ ä
£
Ó Î { x È Ç X ÕLiÀ v *Õ`Ã
Lesson 7-4 Constant Rate of Change and Direct Variation
379
Example 2 (p. 378)
Determine whether a proportional linear relationship exists between the two quantities shown in each of the functions indicated. Explain your reasoning. 3. Exercise 1
(p. 379)
HOMEWORK
HELP
Find the constant rate of change for each linear function and interpret its meaning. 7.
ÕÃÌâi` ÛÌ>ÌÃ
8.
iÀV> À«>iÃ
{ä
xäää
Îx
{äää
Îä
ÃÌ f®
For See Exercises Examples 7–10 1 11–14 2 15–18 3
PHYSICAL SCIENCE The length of a spring stretches directly with the amount of weight attached to it. When a 25-gram weight is attached, a spring stretches 8 centimeters. 5. Write a direct variation equation relating the weight x and the amount of stretch y. 6. Estimate the stretch of the spring when it has a 60-gram weight attached.
ÃÌ>Vi ®
Example 3
4. Exercise 2
Óx Óä
Îäää Óäää £äää
£x ä
£ä
£
Ó
x
Î { x /i ®
È
Ç
n
x £ä £x Óä Óx Îä Îx {ä
ä
->ià f®
9.
vÌ} 7i} ÌÃ
10.
iÌÌ} /ViÌÃ
Y
Y Çä /ViÌÃ Û>>Li
/Ì> 7i} Ì L®
Îx Îä Óx Óä £x £ä x ä
Èä xä {ä Îä Óä £ä
£ Ó Î { x È Ç X ÕLiÀ v À 7i} ÌÃ
ä
£ä Óä Îä {ä xä Èä Çä X /ViÌà -`
Determine whether a proportional linear relationship exists between the two quantities shown in each of the functions indicated. Explain your reasoning. 11. Exercise 7 EXTRA
PRACTICE
See pages 777, 800. Self-Check Quiz at pre-alg.com
12. Exercise 8
13. Exercise 9
14. Exercise 10
FOOD COSTS The cost of cheese varies directly with the number of pounds bought. Suppose 2 pounds cost $8.40. 15. Write an equation that could be used to find the unit cost of cheese. 16. Find the cost of 3.5 pounds of cheese.
380 Chapter 7 Functions and Graphing
CONVERSIONS The number of centimeters in a measure varies directly as the number of inches. 17. Write an equation that could be used to convert inches to centimeters. 18. How many inches is 16.51 centimeters?
H.O.T Problems
19. OPEN ENDED Graph a line that shows a 2-unit increase in y for every 1-unit increase in x. State the rate of change. 20. REASONING Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. A linear relationship that has a constant rate of change is a proportional relationship. 21.
Writing in Math Write about two quantities in real life that have a proportional linear relationship. Describe how you could change the situation to make the relationship between the quantities nonproportional.
22. Which is a true statement about the graph below? A There is not a constant rate of change.
i>} >À«iÌà f£Èä /Ì> ÃÌ
f£{ä
B The two quantities are not proportional.
f£Óä f£ää
C A proportional linear relationship exists.
fnä fÈä ä
D The total cost varies directly with the number of rooms.
£ Ó Î { ÕLiÀ v ,Ã
23. ENTERTAINMENT Admission to a water park is $36 for 3 people, $48 for 4 people, and $60 for 5 people. What is the rate of change, and what does the rate mean in this situation? (Lesson 7-3) Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2) 1 25. y = _ x
24. y = 5 - 3x
26. x - y = 10
2
27. SPACE The table shows how many stars a person can see in the night sky. How many stars can be seen with a small telescope? (Lesson 4-1)
Unaided eye in urban area
3 · 102 stars
Unaided eye in rural area
2 · 103 stars
With binoculars
3 · 104 stars
With small telescope
2 · 106 stars
Source: Kids Discover
PREREQUISITE SKILL Subtract. (Lesson 2-3) 28. -11 - 13
29. 15 - 31
30. -26 - (-26)
31. 9 - (-16)
Lesson 7-4 Constant Rate of Change and Direct Variation
381
CH
APTER
7
Mid-Chapter Quiz Lessons 7-1 through 7-4
Determine whether each relation is a function. Explain. (Lesson 7-1) 1. {(0, 5), (1, 2), (1, -3), (2, 4)} 2. {(-6, 3.5), (-3, 4.0), (0, 4.5), (3, 5.0)} x
9
11
13
17
21
y
7
3
-1
-5
-7
4.
->Û}Ã
>>Vi f®
3.
12. SAVINGS The graph shows Felisa’s and Julian’s savings accounts several weeks after they were opened. Compare the rates of change. (Lesson 7-3)
Y
£ää ä nä Çä Èä xä {ä Îä Óä £ä ä
Y iÃ>
Õ>
£
Ó
5. MULTIPLE CHOICE The relation {(2, 11), (-9, 8), (14, 1), (5, 5)} is NOT a function when which ordered pair is added to the set? (Lesson 7-1)
A (8, -9)
C (0, 0)
B (6, 11)
D (2, 18)
Find four solutions of each equation. Write the solutions as ordered pairs. (Lesson 7-2)
X
,iÌ} 6`i >iÃ
7. y = 5x - 1
fÓx
Graph each equation by plotting ordered pairs. (Lesson 7-2) 8. 3x - y = 7
x
PART TIME JOB For Exercises 13 and 14, use the following information. Ivy’s income varies directly with the number of hours she works. When Ivy works 4 hours she earns $44. (Lesson 7-4) 13. Write an equation that relates hours worked x and amount of pay y. 14. Predict the amount earned after 20 hours of work. 15. Find the constant rate of change for the linear function shown below and interpret its meaning. (Lesson 7-4)
9. x = 2
10. INSECTS The average flea can jump 150 times its own length. This can be represented by the equation y = 150x, where x is a flea’s 1 -inch long length. How far can a flea that is 16 jump? (Lesson 7-2) 11. TRAVEL Find the rate of change for the linear function. (Lesson 7-3) Time (h)
x
0.5
1.5
3
Distance (mi)
y
30
90
180
382 Chapter 7 Functions and Graphing
ÕÌ ,i>}
6. y = 8
{
7iiÃ
X
•
Î
fÓä f£x f£ä fx
ä
£
Ó
Î
{
x
ÕLiÀ v >iÃ
16. MULTIPLE CHOICE Which is a true statement about the graph shown in Exercise 15? (Lesson 7-4) F A proportional linear relationship exists. G The two quantities are not proportional. H There is not a constant rate of change. J The amount remaining varies directly with the number of games.
EXPLORE
7-5
Algebra Lab
It’s All Downhill The steepness, or slope, of a hill can be described by a ratio. vertical change height slope = __ horizontal change length
hill
vertical change
horizontal change
• Use posterboard or a wooden board, tape, and three or more books to make a “hill.”
y x
• Measure the height y and 1 1 inch or _ inch. Record the length x of the hill to the nearest _ 2 4 measurements in a table like the one below. Hill
Height y (in.)
Length x Car Distance (in.) (in.)
y
Slope x
1 2 3
ACTIVITY Step 1 Place a toy car at the top of the hill and let it roll down. Measure the distance from the bottom of the ramp to the back of the car when it stops. Record the distance in the table. Step 2
For the second hill, increase the height by adding one or two more books. Roll the car down and measure the distance it rolls. Record the dimensions of the hill and the distance in the table.
Step 3 Take away two or three books so that hill 3 has the least height. Roll the car down and measure the distance it rolls. Record the dimensions of the hill and the distance in the table. Step 4 Find the slopes of hills 1, 2, and 3 and record the values in the table.
ANALYZE THE RESULTS 1. How did the slope change when the height increased and the length decreased? 2. How did the slope change when the height decreased and the length increased? 3. MAKE A CONJECTURE On which hill would a toy car roll the farthest— 18 4 or _ ? Explain by describing the relationship a hill with slope _ 25 18 between slope and distance traveled. 4. Make a fourth hill. Find its slope and predict the distance a toy car will go when it rolls down the hill. Test your prediction by rolling a car down the hill. Explore 7-5 Algebra Lab: It’s All Downhill
383
7-5
Slope
Main Idea • Find the slope of a line.
New Vocabulary slope
Some roller coasters can make you feel heavier than a shuttle astronaut feels on liftoff. This is because the speed and steepness of the hills increase the effects of gravity. a. Write the rate of change comparing the height of the roller coaster to the length of the drop as a fraction in simplest form.
56 ft
b. Find the rate of change of a hill that has the same length but is 14 feet higher than the hill above. Is this hill steeper or less steep than the original?
42 ft
Slope Slope describes the steepness of a line. It is the ratio of the rise, or the vertical change, to the run, or the horizontal change. rise slope = _ run
← vertical change ← horizontal change
Note that the slope is the same for any two points on a straight line. It represents a constant rate of change.
Use Rise and Run to Find Slope ROADS Find the slope of a road that rises 25 feet for every horizontal change of 80 feet. rise slope = _ run 25 ft =_ 80 ft 5 =_ 16
Write the formula. rise = 25 ft, run = 80 ft
25 ft
Simplify. 80 ft
5 The slope of the road is _ or 0.3125. 16
1. RAMPS What is the slope of a wooden wheelchair ramp that rises 2 inches for every horizontal change of 24 inches? 384 Chapter 7 Functions and Graphing Tony Freeman/PhotoEdit
EXAMPLE
Use a Graph to Find Slope
Find the slope of each line. a.
b.
ÀÕ Î
Y
{] Ó®
È] n®
"
X
ÀÃi È
ÀÃi { Î] {®
x] {® ÀÕ
"
rise _4 slope = _ run =
rise -6 2 _ slope = _ or -_ run = 9
3
2A.
2B.
Y
Ó] Ó®
3
Y "
X
ä] ä®
" X Ó] {® x] ή
Slope The slope m of a line passing through points at (x1, y1) and (x2, y2) is the ratio of the difference in y-coordinates to the corresponding difference in x-coordinates.
Words
Model
y (x 1, y 1) (x 2, y 2) O
y2 - y1
Symbols m = _ x - x , where x2 ≠ x1 2
x
1
Choosing Points • Any two points on a line can be chosen as (x1, y1) and (x2, y2). • The coordinates of both points must be used in the same order. Check: In Example 3a, let (x1, y1) = (5, 3) and let (x2, y2) = (2, 2), then find the slope.
EXAMPLE
Positive and Negative Slopes
Find the slope of each line. a.
b.
y (2, 2)
x
y -y
2 1 m=_ x -x
3-2 m=_ 5-2 _ m= 1 3
1
x
O
(5, 3)
O
2
y (⫺2, 1)
Definition of slope (x1, y1) = (2, 2), (x2, y2) = (5, 3)
(0, ⫺3)
y -y
2 1 m=_ x -x 2
1
Definition of slope
(x1, y1) = (-2, 1), 0 - (-2) (x2, y2) = (0, -3) -4 m=_ or -2 2
-3 - 1 m=_
Find the slope of the line that passes through each pair of points. 3B. C(1, -5), D(8, 3) 3A. A(-4, 3), B(1, 2) Extra Examples at pre-alg.com
Lesson 7-5 Slope
385
EXAMPLE
Zero and Undefined Slopes
Find the slope of each line. a.
b.
y
y (⫺5, 3)
(⫺1, 1)
(3, 1) (⫺5, 0) x
O
y -y
2 1 m=_ x -x 2
1
1-1 m=_
3 - (-1) _ m = 0 or 0 4
O
y -y
x
Definition of slope
2 1 m=_ x -x
(x1, y1) = (-1, 1), (x2, y2) = (3, 1)
0-3 (x1, y1) = (-5, 3), m =_ -5 - (-5) (x2, y2) = (-5, 0)
2
1
-3 m=_
Definition of slope
Division by 0 is undefined.
The slope is undefined.
Find the slope of the line that passes through each pair of points. 4B. G(2, 4), H(2, -1) 4A. E(-1, 7), F(5, 7)
Compare Slopes There are two major hills on a hiking trail. Hill 1 rises 6 feet vertically for every 42-foot run. Hill 2 rises 10 feet vertically for every 98-foot run. Which statement is true? Make a Drawing Whenever possible, make a drawing that displays the given information. Then use the drawing to estimate the answer.
A Hill 1 is steeper than Hill 2.
C Both hills have the same steepness.
B Hill 2 is steeper than Hill 1.
D You cannot find which hill is steeper.
Read the Test Item To compare the steepness of the hills, find the slopes. Solve the Test Item Hill 1
Hill 2
rise slope = _ run
rise slope = _ run
6 ft =_
rise = 6 ft, run = 42 ft 42 ft 1 =_ or about 0.14 7
10 ft =_
rise = 10 ft, run = 98 ft 98 ft 5 =_ or about 0.10 49
0.14 > 0.10, so the first hill is steeper than the second. The answer is A.
5. A home builder has four models with roofs having the dimensions in the table. Which roof is the steepest? F roof A G roof B H roof C J roof D
Personal Tutor at pre-alg.com
386 Chapter 7 Functions and Graphing
Roof A B C D
Length (ft) Height (ft) 15 5 16 4 20 10 21 14
Example 1 (p. 384)
1. CARPENTRY In a stairway, the slope of the handrail is the ratio of the riser to the tread. If the tread is 12 inches long and the riser is 8 inches long, find the slope.
handrail
tread riser
Example 2 (p. 385)
Find the slope of each line. 2.
3.
y
y (⫺1, 2)
(⫺3, 0) O
x
(1, 2)
x
O
(0, ⫺2)
Examples 3, 4
Find the slope of the line that passes through each pair of points.
(pp. 385–386)
4. A(3, 4), B(4, 6)
Example 5
7. MULTIPLE CHOICE Which bike ramp is the steepest? A 1 C 3 B 2 D 4
(p. 386)
HOMEWORK
HELP
For See Exercises Examples 8, 9 1 10–13 2 14–19 3, 4 33, 34 5
5. J(-8, 0), K(-8, 10)
6. X(-7, 0), Y(-1, -5) Bike Ramp 1 2 3 4
Length (ft) 8 4 3 4
Height (ft) 6 10 5 8
8. SKIING Find the slope of a snowboarding beginner hill that decreases 24 feet vertically for every 30-foot horizontal increase. FT
9. HOME REPAIR The bottom of a ladder is placed 4 feet away from a house. It reaches a height of 16 feet on the side of the house. What is the slope of the ladder?
FT
Find the slope of each line. 10.
11.
y
y
(1, 3) (⫺2, 0) O
x
x
O (⫺1, ⫺1) (3, ⫺4)
Lesson 7-5 Slope
387
Find the slope of each line. Y
12.
13.
Y
Î] Ó®
È] £®
X
"
X
" Ó] £®
Î] ή
Find the slope of the line that passes through each pair of points. 14. A(1, -3), B(5, 4) 17. J(-3, 6), K(-5, 9)
15. Y(4, -3), Z(5, -2) 18. N(2, 6), P(-1, 6)
16. S(-9, -4), T(-9, 8) 19. D(5, -1), E(-3, 4)
20. ROLLER COASTERS The first hill of the Texas Giant at Six Flags over Texas drops 137 feet vertically over a horizontal distance of about 103 feet. The Magnum XL-200 at Cedar Point in Ohio drops 195 feet vertically over a horizontal distance of about 113 feet. Which coaster has the steeper slope?
The Kingda Ka roller coaster in Jackson, New Jersey, is the tallest roller coaster in the world, standing at 456 feet. Source: rcdb.com
ANALYZE GRAPHS For Exercises 21–23, use the graph. 21. Which section of the graph shows the smallest increase in attendance? Describe the slope. 22. What happened to the attendance at the amusement park from 2002–2003? Describe the slope of this part of the graph. 23. The attendance for 1997 was 300 million. Describe the slope of a line connecting the data points from 1997 to 1998.
ÕÃiiÌ *>À 6ÃÌÀà ÌÌi`>Vi î
Real-World Link
ÎÎä ÎÓx ÎÓä Σx Σä Îäx Îää ä ½n ½ ½ää ½ä£ ½äÓ ½äÎ ½ä{ 9i>À
3OURCE )NTERNATIONAL !SSOCIATION OF !MUSEMENT 0ARKS AND !TTRACTIONS
Find the slope of the line that passes through each pair of points. 1 _ 1 , 5 1 , X 2_ ,6 25. W 3_
(
24. F(0, 1.6), G(0.5, 2.1) EXTRA
PRACTICE
See pages 778, 800. Self-Check Quiz at pre-alg.com
H.O.T. Problems
2
4
) (
2
)
26. ANALYZE TABLES What is the slope of the line represented by the data in the table? 27. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the slopes of two lines or segments on a graph.
x -1 0 1 2
y -6 -8 -10 -12
1 28. OPEN ENDED Draw a line whose slope is -_ . 4
29. FIND THE ERROR Mike and Chloe are finding the slope of the line that passes through Q(-2, 8) and R(11, 7). Who is correct? Explain your reasoning. Mike 8-7 m=_ -2 - 11
Chloe 7-8 m= _ 11 - 2
30. CHALLENGE The graph of a line goes through the origin (0, 0) and C(a, b). State the slope of this line and explain how it relates to the coordinates of point C. 388 Chapter 7 Functions and Graphing Courtesy of Six Flags Theme Parks
31. REASONING Determine whether the following sentence is true or false. If true, provide an example. If false, provide a counterexample. As the constant of variation increases in a direct variation, the slope of the graph becomes steeper. 32.
Writing in Math Explain how slope is used to describe roller coasters. Include a description of slope and an explanation of how changes in rise or run affect the steepness of a roller coaster.
ià /À>Ûii`
{n 33. The graph {ä shows the ÎÓ distance traveled Ó{ by Ebony and £È Rocco during the n first five hours of a seven-hour ä long bicycle ride. Find the true statement.
LÞ ,VV
34. The table shows attendance figures for the Atlanta Falcons football team for three years. Find the slope of a line connecting the years 2001 and 2003 on the graph. Year 2003 2002 2001
£ Ó Î { x È ÕLiÀ v ÕÀÃ
A Rocco’s speed was 6 mph.
Attendance 563,676 550,974 451,333
Source: atlantafalcons.com
B Ebony’s speed was 16 mph. C Rocco traveled a total of 30 miles.
F 12,702
H 99,641
G 56,171.5
J 112,343
D Ebony’s traveled a total of 40 miles.
ÀViÃ
ÀVÕviÀiVi V®
35. Determine whether a proportional linear relationship exists between the two quantities shown in the graph. Explain your reasoning. (Lesson 7-4)
Îä Óä £x £ä x ä
36. FRUIT The table at the right shows the cost of pears and oranges. Compare the rates of change. (Lesson 7-3)
Solve each equation. (Lesson 3-2) 37. -123 = x - 183 38. -205 + t = -118
PREREQUISITE SKILL Solve each equation for y. (Lesson 7-2) 40. x + y = 6
41. 3x + y = 1
42. -x + 5y = 10
Y
Óx
£
Ó Î { x X ,>`ÕÃ V®
Cost y Weight (lb) x
Pears
Oranges
$0
$0
2
$2.40
$2.20
5
$6
$5.50
39. -350 + z = 125
y 43. _ - 7x = -5 2
Lesson 7-5 Slope
389
EXTEND
7-5
Graphing Calculator Lab
Slope and Rate of Change
In this activity, you will investigate the relationship between slope and rate of change.
• Attach the force sensor to the graphing calculator. Place the sensor in a ring stand as shown. • Make a small hole in the bottom of a paper cup. Straighten a paper clip and use it to create a handle to hang the cup on the force sensor. Place another cup on the floor below. • Set the device to collect data 100 times at intervals of 0.1 second.
ACTIVITY Step 1 Hold your finger over the hole in the cup. Fill the cup with water. Step 2 Begin collecting data as you begin to allow the water to drain. Step 3 Make the hole in the cup larger. Then repeat Steps 1 and 2 for a second trial.
ANALYZE THE RESULTS 1. Use the calculator to create a graph of the data for Trial 1. The graph will show the weight of the cup y as a function of time x. Describe the graph. 2. Create the graph for Trial 2. Compare the steepness of the two graphs. Which has a greater slope? 3. What happens as the time increases? 4. Did the cup empty at a faster rate in Trial 1 or Trial 2? Explain. 5. Describe the relationship between slope and the rate at which the cup was emptied. 6. MAKE A CONJECTURE What would a graph look like if you emptied a cup using a hole half the size of the original hole? twice the size of the second hole? Explain. 7. Water is emptied at a constant rate from containers shaped like the ones shown below. Draw a graph of the water level in each of the containers as a function of time. a.
b.
390 Chapter 7 Functions and Graphing Horizons Companies
c.
7-6
Slope-Intercept Form
Main Ideas • Determine slopes and y-intercepts of lines. • Graph linear equations using the slope and y-intercept.
New Vocabulary y-intercept slope-intercept form
A landscaping company charges a $20 fee to mow a lawn plus $8 per hour. The equation y = 8x + 20 represents this situation where x is the number of hours it takes to mow the lawn and y is the total cost of mowing the lawn.
Number of Hours, x 1 2 3
Total Cost, y
a. Copy and complete the table to find the total cost of mowing the lawn. b. Use the table to graph the equation. In which quadrant does the graph lie? Explain. c. Find the y-coordinate of the point where the graph crosses the y-axis and the slope of the line. How are they related to the equation?
Slope and y-Intercept The
y
y-intercept is the y-coordinate of a point where the graph crosses the y-axis. An equation with a y-intercept that is not 0 represents a nonproportional relationship. The equation y = 2x + 1 is written in the form y = mx + b, where m is the slope and b is the y-intercept. This is called slope-intercept form.
Reading Math Different Forms Both equations below are written in slope-intercept form. y = x + (-2) y=x-2
y 2x 1 x
O
slope 2 y-intercept 1
y = mx + b slope
EXAMPLE
y-intercept
Find the Slope and y-Intercept
3 State the slope and the y-intercept of the graph of y = _ x - 7. 5
3 y=_ x-7
Write the original equation.
3 y = _x + (-7)
Write the equation in the form y = mx + b.
↑ ↑ y = mx + b
3 m=_ , b = -7
5 5
5
3 The slope of the graph is _ , and the y-intercept is -7. 5
State the slope and the y-intercept of the graph of each equation. 1A. y = 8x + 6 1B. y = x - 3 Extra Examples at pre-alg.com
Lesson 7-6 Slope-Intercept Form
391
EXAMPLE BrainPOP® pre-alg.com
Write an Equation in Slope-Intercept Form
State the slope and the y-intercept of the graph of 5x + y = 3. 5x + y = 3
Write the original equation.
5x + y - 5x = 3 - 5x
Subtract 5x from each side.
y = -5x + 3 Simplify and write in slope-intercept form. The slope of the graph is -5, and the y-intercept is 3.
State the slope and the y-intercept of the graph of each equation. 1 2A. -9x + y = -5 2B. y - 6 = _x 2
Graph Equations You can use the slope-intercept form of an equation to graph a line.
EXAMPLE
Graph an Equation
1 Graph y = -_ x - 4 using the slope and y-intercept. 2
Step 1
1 Find the slope and y-intercept. m = -_
Step 2
Graph the y-intercept point at (0, -4).
Step 3
2
b = -4 y
-1 1 Write the slope -_ as _ . Use it to locate a 2 2 second point on the line. -1 ← change in y: down 1 unit m=_ 2
← change in x: right 2 units
down 1 unit
Another point on the line is at (2, -5). Step 4
x
O
(0, 4) (2, 5) right 2 units
Draw a line through the two points.
1 3. Graph y = _ x + 1 using the slope and y-intercept. 3
For more information, go to pre-alg.com.
a. Graph the equation. First, find the slope and the y-intercept. slope = 12 y-intercept = 24 Plot the point at (0, 24). Then go up 12 and right 1. Connect these points.
ÃÌ f®
Real-World Career Business Owner Business owners must understand the factors that affect cost and profit. Graphs are a useful way for them to display this information.
BUSINESS A T-shirt company charges a design fee of $24 for a pattern and then sells the shirts for $12 each. The total cost y can be represented by y = 12x + 24, where x represents the number of T-shirts. n{ ÇÓ Èä {n ÎÈ Ó{ £Ó
Y
b. Describe what the y-intercept and the X ä £ Ó Î { x È Ç slope represent. ÕLiÀ v /- ÀÌÃ The y-intercept 24 represents the design fee. The slope 12 represents the cost per T-shirt, which is the rate of change. Since it is not reasonable for the number of T-shirts or the cost to be negative, the graph is in Quadrant I only.
392 Chapter 7 Functions and Graphing Cris Haigh/Getty Images
4. WRITING Tim has written 30 pages of a novel. He plans to write 12 pages per week until he is finished. The total number of pages written y can be represented by y = 12x + 30, where x represents the number of weeks. Graph the equation. Describe what the y-intercept and the slope represent. Personal Tutor at pre-alg.com
Examples 1, 2 (pp. 391–392)
Example 3 (p. 392)
State the slope and the y-intercept of the graph of each equation. 1. y = x + 8
Example 4
HOMEWORK
HELP
For See Exercises Examples 9–14 1, 2 15–23 3 24–27 4
3. x + 3y = 6
Graph each equation using the slope and y-intercept. 1 4. y = _ x+1 4
(pp. 392–393)
2. x + y = 2
5. 3x + y = 2
6. x - 2y = 4
BUSINESS Mrs. Allison charges $25 for a basic cake that serves 12 people. A larger cake costs an additional $1.50 per serving. The total cost can be given by y = 1.5x + 25, where x represents the number of additional slices. 7. Graph the equation. 8. Explain what the y-intercept and the slope represent.
State the slope and the y-intercept of the graph of each equation. 9. y = x + 2 12. 2x + y = -3
10. y = 2x - 4
11. x + y = -3
13. 5x + 4y = 20
14. y = 4
Graph each equation using the slope and y-intercept. 15. y = x + 5
16. y = -x + 6
17. y = 2x - 3
3 18. y = _ x+2 4
19. x + y = -3
20. x + y = 0
21. -2x + y = -1
22. 5x + y = -3
1 23. y + _ x=1 5
AUTOMOBILES For Exercises 24 and 25, use the following information. To replace a set of brakes, an auto mechanic charges $40 for parts plus $50 per hour. The total cost y can be given by y = 50x + 40 for x hours. 24. Graph the equation using the slope and y-intercept. 25. State the slope and y-intercept of the graph of the equation and describe what they represent.
EXTRA
PRACTICE
See pages 778, 800. Self-Check Quiz at pre-alg.com
HANG GLIDING For Exercises 26–28, use the following information. The altitude in feet y of a hang glider who is slowly landing can be given by y = 300 - 50x, where x represents the time in minutes. 26. Graph the equation using the slope and y-intercept. 27. State the slope and y-intercept of the graph of the equation and describe what they represent. 28. The x-intercept is the x-coordinate of a point where a graph crosses the x-axis. Name the x-intercept and describe what it represents. Graph each equation using the slope and y-intercept. 29. x - 3y = -6
30. 2x + 3y = 12
31. y = -3
Lesson 7-6 Slope-Intercept Form
393
H.O.T. Problems
32. OPEN ENDED Draw the graph of a line that has a y-intercept but no x-intercept. What is the slope of the line? 33. FIND THE ERROR Carlotta and Alex are finding the slope and y-intercept of x + 2y = 8. Who is correct? Explain your reasoning. Alex slope = -_1 2 y-intercept = 4
Carlotta slope = 2 y-intercept = 8
34. CHALLENGE What is the x-intercept of the graph of y = mx + b? Explain how you know. 35.
Writing in Math
How can knowing the slope and y-intercept help you graph an equation? Include an explanation of how to write an equation for a line if you know the slope and y-intercept.
36. Which best represents the graph of y = 3x - 1? A
Î Ó £
Y
ÎÓ£ " £ Ó Î
B
Î Ó £
Y Î Ó £
C
ÎÓ£ " £ Ó Î
£ Ó ÎX
Y
D
Î Ó £ ÎÓ£ " £ Ó Î
ÎÓ£ "£ Ó Î X £ Ó Î
£ Ó ÎX
Y
£ Ó ÎX
37. CARS The cost of gas varies directly with the number of gallons bought. Marty bought 18 gallons of gas for $49.50. Write an equation that could be used to find the unit cost of a gallon of gas. Then find the unit cost. (Lesson 7-5) 38. BIRDSEED Find the constant rate of change for the linear function in the table at the right and interpret its meaning. (Lesson 7-4)
PREREQUISITE SKILL Simplify. (Lesson 1-2) 39. 2(18) - 1
40. (-2 - 4) ÷ 10
394 Chapter 7 Functions and Graphing
41. -1(6) + 8
Amount of Birdseed (lb) x 4 8 12
42. 5 - 8(-3)
Total Cost ($) y 11.20 22.40 33.60
EXTEND
7-6
Graphing Calculator Lab
The Family of Linear Graphs
A graphing calculator is a valuable tool when investigating characteristics of linear functions. Before graphing, you must create a viewing window that shows both the x- and y-intercepts of the graph of a function. You can use the standard viewing window [-10, 10] scl: 1 by [-10, 10] scl: 1 or set your own minimum and maximum values for the axes and the scale factor by using the WINDOW option.
The tick marks on the x scale and on the y scale are 1 unit apart.
[-10, 10] scl: 1 by [-10, 10] scl: 1
You can use a TI-83/84 Plus graphing The x-axis goes calculator to enter several functions and from -10 to 10. graph them at the same time on the same screen. This is useful when studying a family of functions. The family of linear functions has the parent function y = x.
The y-axis goes from -10 to 10.
ACTIVITY 1 Graph y = 3x - 2 and y = 3x + 4 in the standard viewing window and describe how the graphs are related. Step 1 Graph y = 3x + 4 in the standard viewing window. • Clear any existing equations from the Y= list. KEYSTROKES:
y ⫽ 3x ⫹ 4
CLEAR
• Enter the equation and graph. KEYSTROKES:
3 X,T,,n
4 ZOOM 6
Step 2 Graph y = 3x - 2. • Enter the function y = 3x – 2 as Y2 with y = 3x + 4 already existing as Y1. KEYSTROKES:
3 X,T,,n
2
• Graph both functions in the standard viewing window. KEYSTROKES:
ZOOM 6
The first function graphed is Y1 or y = 3x + 4. The second function graphed is Y2 or y = 3x - 2. Press TRACE . Move along each function using the right and left arrow keys. Move from one function to another using the up and down arrow keys. The graphs have the same slope, 3, but different y-intercepts at 4 and -2. Extend 7-6 Graphing Calculator Lab: The Family of Linear Graphs
395
EXERCISES Graph y = 2x - 5, y = 2x - 1, and y = 2x + 7. 1. Compare and contrast the graphs. 2. How does adding or subtracting a constant c from a linear function affect its graph? 3. Write an equation of a line whose graph is parallel to y = 3x - 5, but is shifted up 7 units. 4. Write an equation of the line that is parallel to y = 3x - 5 and passes through the origin. 5. Four functions with a slope of 1 are graphed in the standard viewing window, as shown at the right. Write an equation for each, beginning with the left most graph.
[-10, 10] scl:1 by [-10, 10] scl:1
_
_
Clear all functions from the Y= menu and graph y = 1 x, y = 3 x, y = x, and 3 4 y = 4x in the standard viewing window. 6. How does the steepness of a line change as the coefficient for x increases? 7. Without graphing, determine whether the graph of y = 0.4x or the graph of y = 1.4x has a steeper slope. Explain. Clear all functions from the Y= menu and graph y = -4x and y = 4x. 8. How are these two graphs different? 9. How does the sign of the coefficient of x affect the slope of a line? 10. Describe the similarities and differences between the graph of y = 2x - 3 and the graph of each equation listed below. a. y = 2x + 3 b. y = -2x – 3 c. y = 0.5x + 3 11. Write an equation of a line whose graph lies between the graphs of y = -3x and y = -6x. For Exercises 12–14, use the following information. A garden center charges $75 per cubic yard for topsoil. The delivery fee is $25. 12. Describe the change in the graph of the situation if the delivery fee is changed to $35. 13. How does the graph of the situation change if the price of a cubic yard of topsoil is increased to $80? 14. What are the prices of a cubic yard of topsoil and delivery if the graph has slope 70 and y-intercept 40? 396 Chapter 7 Functions and Graphing
7-7
Writing Linear Equations
Main Idea • Write equations given the slope and y-intercept, a graph, a table, or two points.
You can determine the approximate outside temperature by counting the chirps of crickets, as shown in the table.
Number of Chirps in 15 Seconds
Temperature (°F)
40
5
45
10
50
15
55
20
60
a. Graph the ordered pairs (chirps, temperature). Draw a line through the points. b. Find the slope and the y-intercept of the line. What do these values represent?
c. Write an equation in the form y = mx + b for the line. Then translate the equation into a sentence.
Write Equations There are many different methods for writing linear equations. If you know the slope and y-intercept, you can write the equation of a line by substituting these values in y = mx + b.
EXAMPLE
Write Equations From Slope and y-Intercept
Write an equation in slope-intercept form for each line. a. slope = 4, y-intercept = -8 y = mx + b
b. slope = 0, y-intercept = 5 y = mx + b
Slope-intercept form
y = 4x + (-8) Substitute
y = 0x + 5
Substitute
y = 4x - 8
y=5
Simplify.
Slope-intercept form
Simplify.
1 1A. slope = -_ , y-intercept = 0 2
EXAMPLE
1 1B. slope = 2, y-intercept = -_ 3
Write an Equation From a Graph y
Write an equation in slope-intercept form for the line graphed. Check Equation To check, choose another point on the line and substitute its coordinates for x and y in the equation.
The y-intercept is 1. From (0, 1), you can go down 3 units and right 1 unit to another point -3 , or -3. on the line. So, the slope is _ y = mx + b
1 Slope = intercept form
y = -3x + 1 Replace m with -3 and b with 1.
y -intercept
down 3 units
O
x
right 1 unit
Lesson 7-7 Writing Linear Equations
397
2. Write an equation in slope-intercept form for the line graphed.
Y
X
"
EXAMPLE
Write an Equation to Make a Prediction
EARTH SCIENCE On a summer day, the temperature at altitude 0 meters, or sea level, is 30°C. The temperature decreases 2°C for every 305 meters increase in altitude. Predict the temperature for an altitude of 2000 meters.
Use a Table Translate the words into a table of values to help clarify the meaning of the slope. For every increase of 305 meters in altitude, the temperature decreases by 2°C. Alt. (m)
Explore You know the rate of change of temperature to altitude (slope) and the temperature at sea level (y-intercept). Make a table of ordered pairs.
30
305
28
610
26
30
305
28
610
26
Plan
Write an equation to show the relationship between altitude x and temperature y. Then, substitute the altitude of 2000 meters into the equation to find the temperature.
Solve
Step 1
Temp. (°C)
Altitude, Temperature y (°C), x
Step 2 Find the y-intercept b.
Find the slope m. change in y m=_ change in x -2 = 305
≈ -0.007
← ← ← decrease of -2°C ← increase of 305 m
change in temperature __ change in altitude
Simplify.
Step 3
(x, y) = (altitude, temperature) = (0, b) When the altitude is 0, or sea level, the temperature is 30°C. So, the y-intercept is 30.
Write the equation. y = mx + b Slope-intercept form ≈ -0.007x + 30 Replace m with -0.007 and b with 30. Step 4 Substitute the altitude of 2000 meters. Write the equation. y = -0.007x + 30 = -0.007(2000) + 30 Replace x with 2000. ≈ 17 Simplify.
So, at an altitude of 2000 meters, the temperature is about 17°C. Check
As you go up, the temperature drops about 2°C for every 300 meters. Since 2000 ÷ 300 is about 7, at 2000 meters the temperature will drop about 7 × 2°C or 14°C. If the temperature is 30°C at 0 meters, the temperature at 2000 meters is about 30 - 14 or about 16°C. So the answer is reasonable.
398 Chapter 7 Functions and Graphing
Extra Examples at pre-alg.com
3. PIANO LESSONS The cost of 7 half-hour piano lessons is $151. The cost of 11 half-hour lessons is $223. Write a linear equation that shows the cost y for x half-hour lessons. Then use the equation to find the cost of 3 half-hour lessons. Personal Tutor at pre-alg.com
You can also write an equation for a line if you know the coordinates of two points on a line.
EXAMPLE
Write an Equation Given Two Points
Write an equation for the line that passes through (-2, 5) and (2, 1). Step 1
Find the slope m. y2 - y1 m=_ x -x 2
Definition of slope
1
5-1 =_ or -1 (x1, y1) = (-2, 5), (x2, y2) = (2, 1) -2 - 2
Step 2
y = mx + b 5 = -1(-2) + b 3=b
Check Equation To check, substitute the coordinates of the other point into the equation.
Find the y-intercept b. Use the slope and the coordinates of either point.
Step 3
Slope-intercept form Replace (x, y) with (-2, 5) and m with -1. Simplify.
Substitute the slope and y-intercept. y = mx + b Slope-intercept form y = -1x + 3 Replace m with -1 and b with 3. y = -x + 3 Simplify.
y = -x + 3 (1) -(2) + 3 1 -2 + 3 1=1
4. Write an equation for the line that passes through (5, 1) and (8, -2).
EXAMPLE
Write an Equation From a Table
Use the table of values to write an equation in slope-intercept form. Step 1 Find the slope m. Use the coordinates of any two points. y2 - y1 m=_ x -x 2
Definition of slope
1
-2 - 6 4 m=_ or -_ (x1, y1) = (-5, 6), (x2, y2) = (5, -2)
Alternate Strategy
Step 2
y = mx + b
Slope-intercept form Replace (x, y) with (-5, 6) and m with -45. Simplify.
x
y
-5
6
2=b
2
y-intercept = 2
-5
6
5
-2
10
-6
15
-10
Find the y-intercept b. Use the slope and the coordinates of any point. 6 = -45(-5) + b
Step 3
y
5
5 - (-5)
If a table includes the y-intercept, simply use this value and the slope to write an equation.
x
Substitute the slope and y-intercept. y = mx + b
Slope-intercept form
y = -45 x + 2
4 Replace m with -_ and b with 2. 5
Lesson 7-7 Writing Linear Equations
399
5. Write an equation in slope-intercept form to represent the table of values shown below.
Example 1 (p. 397)
-6 -3
3
y
-1
2
Write an equation in slope-intercept form for each line. 1. slope = 1, y-intercept = 1
2. slope = 0, y-intercept = -7
3.
4.
2
Example 2
x
y
y
(pp. 397–398) x
O x O
Example 3 (pp. 398–399)
Example 4 (p. 399)
PICNICS It costs $50 plus $10 per hour to rent a park pavilion. 5. Write an equation in slope-intercept form that shows the cost y for renting the pavilion for x hours. 6. Find the cost of renting the pavilion for 8 hours. Write an equation in slope-intercept form for the line passing through each pair of points. 7. (2, 2) and (4, 3)
Example 5 (pp. 399–400)
HOMEWORK
HELP
For See Exercises Examples 10–15 1 16, 17 3 18–23 2 24–29 4 30, 31 5
8. (3, -4) and (-1, 4)
9. Write an equation in slope-intercept form to represent the table of values.
x
-4
4
8
y
-4
-1
2
5
Write an equation in slope-intercept form for each line. 10. slope = 2, y-intercept = 6
11. slope = -4, y-intercept = 1
12. slope = 0, y-intercept = 5
13. slope = 1, y-intercept = -2
1 14. slope = -_ , y-intercept = 8 3
2 15. slope = _ , y-intercept = 0 5
SOUND For Exercises 16 and 17, use the table that shows the distance that a rip current travels through the ocean. 16. Write an equation in slope-intercept form to represent the data in the table. Describe what the slope means. 17. Estimate how far the rip current travels in one minute.
400 Chapter 7 Functions and Graphing
Times (s)
Distance (ft)
x
y
1
2.4
2
4.8
3
7.2
Write an equation in slope-intercept form for each line. 18.
19.
y
20.
y
x
O
x
22.
y
23.
y
x
O
Coyotes communicate by using different barks and howls. Because of the way in which sound travels, a coyote is usually not in the area from which the sound seems to be coming. Source: livingdesert.org
EXTRA
PRACTIICE
y
x
O
Real-World Link
x
O
O
21.
y
x
O
Write an equation in slope-intercept form for the line passing through each pair of points. 24. (-2, -1) and (1, 2)
25. (-4, 3) and (4, -1)
26. (0, 0) and (-1, 1)
27. (4, 2) and (-8, -16)
28. (8, 7) and (-9, 7)
29. (5, -6) and (3, 2)
Write an equation in slope-intercept form for each table of values. 30.
x
-1
1
2
y
-7
-3
1
5
31.
x
-3
-1
1
3
y
7
5
3
1
SOUND For Exercises 32 and 33, use the table that shows the distance that sound travels through dry air at 0°C. 32. Write an equation in slope-intercept form to represent the data in the table. Describe what the slope means. 33. Estimate the number of miles that sound travels through dry air in one minute.
Times (s)
Distance (ft)
x
y
1
1088
2
2176
3
3264
See pages 778, 800. Self-Check Quiz at pre-alg.com
H.O.T. Problems
34. COMPUTERS A computer repair company charges a fee and an hourly charge. After two hours the repair bill is $110, and after three hours it is $150. How much would it cost for 1.5 hours of work? 35. OPEN ENDED Choose a slope and y-intercept. Write an equation and then graph the line. 36. SELECT A TOOL Mr. Awan has budgeted $860 to have his dining room painted. The estimated cost for materials is $100. The painter charges $35 per hour and estimates that the work will take about 20 hours to complete. Which of the following tools might Mr. Awan use to determine whether he has budgeted enough money to paint the dining room? Justify your choice. Then use the tool to solve the problem. draw a model
paper/pencil
calculator
Lesson 7-7 Writing Linear Equations Gail Shumway/Getty Images
401
CHALLENGE A CD player has a pre-sale price of $c. Kim buys it at a 30% discount and pays 6% sales tax. After a few months, she sells it for $d, which was 50% of what she paid originally. 37. Express d as a function of c. 38. How much did Kim sell it for if the pre-sale price was $50? 39.
Writing in Math Explain how you can model data with a linear equation and how to find the y-intercept and slope by using a table.
41. The graphs show how much a video store pays for three different movies. Which equation is NOT represented by one of the graphs? fä fnä
f£ää
fÇä
fä
fÈä
fnä
fxä
ÃÌ
ÕÌ Ài> "ÜiÃ
40. Lorena borrowed $100 from her father and plans to pay him back at a rate of $10 per week. The graph shows the amount Lorena owes her father. Find the equation that represents this relationship.
fÇä
-OVIE " -OVIE !
f{ä
fÈä
fÎä
fxä
fÓä
f{ä
f£ä
fÎä
fä
-OVIE #
ä
fÓä
£
Ó Î { x È Ç ÕLiÀ v ÛiÃ
f£ä
n
F y = 10x
fä ä
£
Ó
Î { x È Ç n ÕLiÀ v 7iiÃ
£ä
G y = 25x
A y = 10x - 100
C y = 100 - 10x
H y = 50x
B y = 10x - 90
D y = -100 - 10x
J y = 60x
State the slope and the y-intercept for the graph of each equation. (Lesson 7-6) 42. y = 6x + 7
43. y = -x + 4
44. -3x + y = -2
45. CONSTRUCTION Find the slope of the road that rises 48 feet for every 144 feet measured horizontally. (Lesson 7-5) Write each number in scientific notation. (Lesson 4-7) 46. 345,000
47. 1,680,000
48. 0.00072
49. 0.001
PREREQUISITE SKILL (Lesson 1-7) 50. State whether a scatter plot containing the following set of points would show a positive, negative, or no relationship. 402 Chapter 7 Functions and Graphing
x
2
4
5
4
3
6
y
15
20
36
44
32
30
50
7-8
Prediction Equations
• Draw lines of fit for sets of data. • Use lines of fit to make predictions about data.
New Vocabulary line of fit
The scatter plot shows the number of years people in the United States are expected to live, according to the year they were born. a. Use the line drawn through the points to predict the life expectancy of a person born in 2020. b. What are some limitations in using a line to predict life expectancy?
Life Expectancy (yr)
Main Ideas
85 80 75 70 65 60 55 50 45 40 0
y
1920 1940 1960 1980 2000 Year
x
Source: The World Almanac
Lines of Fit When real-life data are collected, the points graphed usually do not form a straight line, but may approximate a linear relationship. A line of fit can be used to show such a relationship. A line of fit is a line that is very close to most of the data points.
EXAMPLE
Make Predictions from a Line of Fit
MONEY The table shows the changes in the number of college-bound students taking the ACT.
Labeling In Example 1, the x-axis could have been labeled “Years Since 1991” to simplify the graph and the prediction equation.
£Óää ££ää £äää ää
Number (thousands)
1991 1995 1998 2000 2002 2004
796 945 995 1065 1116 1171
Source: ACT, Inc.
nää ½ä
½ää 9i>À
½£ä
b. Use the line of fit to predict the number of students taking the ACT in 2015. Extend the line so that you can find the y-value for an x-value of 2015. The y-value for 2015 is about 1480. So, the number of students taking the ACT is approximately 1480.
ÕLiÀ v -ÌÕ`iÌÃ Ì ÕÃ>`î
ÕLiÀ v -ÌÕ`iÌÃ Ì ÕÃ>`î
a. Make a scatter plot and draw a line of fit for the data.
Year
£xää £{ää £Îää £Óää ££ää £äää ää nää ä ½ä
½ää ½£ä 9i>À
Lesson 7-8 Prediction Equations
½Óä
403
1. ENTERTAINMENT Make a scatter plot of the percent of U.S. households that have a digital video camera and draw a line of fit. Predict the percent of U.S. households with a digital video camera in 2015. Year
1999
2000
2001
2002
2003
2004
2005
% of U.S. Households
3%
7%
10%
14%
17%
19%
21%
Source: Parks Associates
Personal Tutor at pre-alg.com
Prediction Equations You can also make predictions from the equation of a line of fit.
EXAMPLE
Make Predictions from an Equation
SWIMMING The scatter plot shows the winning Olympic times in the women’s 800-meter freestyle event from 1968 through 2004.
There may be general trends in sets of data. However, not every data point may follow the trend exactly.
y 560
a. Write an equation in slope-intercept form for the line of fit that is drawn. Step 1 First, select two points on the line and find the slope. We have chosen (1980, 525) and (1992, 500). Notice that they are not original data points. y2 - y1 m=_ x -x 2
1
Time (s)
Trends
Women’s 800-Meter Freestyle Event
(1980, 525)
540 520 500
(1992, 500) 0 ’68
Definition of slope
525 - 500 =_ 1980 - 1992
(x1, y1) = (1992, 500), (x2, y2) = (1980, 525)
≈ -2.1
Simplify.
’76
’84 Year
’92
’00
Source: The World Almanac
Step 2 Next, find the y-intercept. y = mx + b
Slope-intercept form
525 = -2.1(1980) + b Replace (x, y) with (1980, 525) and m with -2.1. 4683 ≈ b
Simplify.
Step 3 Write the equation.
Lines of fit can help make predictions about recreational activities. Visit pre-alg.com to continue work on your project.
y = mx + b
Slope-intercept form
y = -2.1x + 4683
Replace m with -2.1 and b with 4683.
b. Predict the winning time in the women’s 800-meter freestyle event in the year 2012. y = -2.1x + 4683
Write the equation of the line of fit.
= -2.1(2012) + 4683
Replace x with 2012.
= 457.8
Simplify.
A prediction for the winning time in the year 2012 is approximately 457.8 seconds or 7 minutes, 37.8 seconds.
404 Chapter 7 Functions and Graphing
Extra Examples at pre-alg.com
x
Time (s)
’68 ’72 ’76 ’80 ’84 ’88 ’92 ’96 ’00 ’04
2. SWIMMING Write an equation in slope-intercept form for the line of fit that is drawn. Predict the winning time in the men’s 100-meter butterfly in 2012.
56 55 54 53 52 51
Year Source: The World Almanac
Example 1 (p. 403)
NEWSPAPERS For Exercises 1 and 2, use the table that shows the number of Sunday newspapers in the U.S. 1. Make a scatter plot and draw a line of fit. 2. Use the line of fit you drew in Exercise 1 to predict the number of Sunday newspapers in the U.S. in 2010.
Year 1998 1999 2000 2001 2002 2003
Sunday Newspapers 898 905 917 913 913 917
Example 2 (p. 404)
SPENDING For Exercises 3 and 4, use the line of fit drawn that shows the billions of dollars spent by travelers in the United States. 3. Write an equation in slope-intercept form for the line of fit. 4. Use the equation to predict how much money travelers will spend in 2008.
Amount ($ billions)
Source: Statistical Abstract of the U.S.
600 500 400 300 200 100
(4, 479) (6, 502.5)
1 2 3 4 5 6 7 Years Since 1997
Source: Travel Industry Association of America
HOMEWORK
HELP
For See Exercises Examples 5, 6, 10, 11 1 7–9, 12–15 2
ENTERTAINMENT For Exercises 5 and 6, use the table that shows the number of movie tickets sold in the United States. Year
1998
1999
2000
2001
2002
2003
2004
Tickets Sold (millions)
1481
1465
1421
1487
1578
1523
1507
Source: boxofficemojo.com
5. Make a scatter plot and draw a line of fit. 6. Use the line of fit to predict movie attendance in 2010. PRESSURE For Exercises 7–9, use the table that shows the approximate barometric pressure at various altitudes. 7. Make a scatter plot of the data and draw a line of fit. 8. Write an equation for the line of fit you drew in Exercise 7. Use it to estimate the barometric pressure at 60,000 feet. Is the estimation reasonable? Explain. 9. Do you think that a line is the best model for this data? Explain.
Altitude (ft) 0 5000 10,000 20,000 30,000 40,000 50,000
Barometric Pressure (in. mercury) 30 25 21 14 9 6 3
Source: New York Public Library Science Desk Reference
Lesson 7-8 Prediction Equations
405
POLE VAULTING For Exercises 10 and 11, use the table that shows the men’s winning Olympic pole vault heights to the nearest inch. 10. Make a scatter plot and draw a line of fit. 11. Use the line of fit to predict the winning pole vault height in the 2008 Olympics. Year
1976
1980
1984
1988
1992
1996
2000
2004
Height (in)
217
228
226
232
228
233
232
234
Source: The World Almanac
EARTH SCIENCE For Exercises 12–15, use the table that shows the latitude and the average temperature in July for five cities in the United States. Real-World Link In 1964, thirteen competitors broke or equaled the previous Olympic pole vault record a total of 36 times. This was due to the new fiberglass pole.
City
Source: Chance
Latitude ( N)
Average July High Temperature ( F)
Chicago, IL
41
73
Dallas, TX
32
85
Denver, CO
39
74
New York, NY
40
77
Duluth, MN
46
66
Fresno, CA
37
97
Source: The World Almanac
EXTRA
PRACTIICE
See pages 779, 800. Self-Check Quiz at pre-alg.com
H.O.T. Problems
12. Make a scatter plot of the data and draw a line of fit. 13. Describe the relationship between latitude and temperature shown by the graph. 14. Write an equation for the line of fit you drew in Exercise 12. 15. Use the equation in Exercise 14 to estimate the average July temperature for a location with latitude 50° north. Round to the nearest degree Fahrenheit. 16. OPEN ENDED Make a scatter plot with at least ten points that appear to be somewhat linear. Draw two different lines that could approximate the data. 17. CHALLENGE The table at the right shows the percent of public schools in the United States with Internet access. Suppose you use (Year, Percent of Schools) to write a linear equation describing the data. Then you use (Years Since 1995, Percent of Schools) to write an equation. Is the slope or y-intercept of the graphs of the equations the same? Explain.
Year
Years Since 1995
Percent of Schools
1995
50
1997
2
78
1999
4
95
2001
6
99
2003
8
100
Source: National Center for Education Statistics
18.
Writing in Math Use the information about life expectancy on page 403 to explain how a line can be used to predict life expectancy for future generations. Include a description of a line of fit and an explanation of how lines can represent sets of data that are not exactly linear.
406 Chapter 7 Functions and Graphing Michael Steele/Getty Images
A The scoring average of the points leader increased over time. B The scoring average of the points leader decreased over time.
-VÀ} i>`iÀà Óΰx Óΰä *Ìà «iÀ >i
19. The scatter plot at the right shows the scoring average of the WNBA points leader from 1997 through 2004. Which statement best describes the relationship on the scatter plot?
ÓÓ°ä Ó£°x Ó£°ä Óä°x Óä°ä
C The scoring average of the points leader remained the same over time. D The scoring average of the points leader could not be determined over time.
ÓÓ°x
ä
¼Ç ¼n ¼ ¼ää ¼ä£ ¼äÓ ¼äÎ ¼ä{ 9i>À
-ÕÀVi\ ÜL>°V
Write an equation in slope-intercept form for each line. (Lesson 7-7) 20. slope = 3, y-intercept = 5
21. slope = -2, y-intercept = 2
22.
23.
y
y x
O
O
x
Graph each equation using the slope and y-intercept. (Lesson 7-6) 24. y = x - 2
1 26. y = _ x
25. y = -x + 3
2
27. ENTERTAINMENT In 2005, a movie actor was to pay 10 percent commission to his management company for work negotiated on his behalf. The commission amount totaled $660,000. How much money did he earn from his movies? (Lesson 6-8) Solve each proportion. (Lesson 6-2) 16 a 28. _ =_ 3
5 15 29. _ =_ x 10
24
n 2 30. _ =_ 16
36
8 12 and y = _ . Write in simplest form. (Lesson 5-3) 31. Evaluate xy if x = _ 9
30
1 GEOGRAPHY Africa makes up _ of all the land on Earth. Use the 5
table to find the fraction of Earth’s land that is made up by other continents. Write each fraction in simplest form. (Lesson 5-2)
Continent
Decimal Portion of Earth’s Land
Antarctica
0.095
32. Antarctica
33. Asia
Asia
0.295
34. Europe
35. North America
Europe
0.07
North America
0.16
Source: Incredible Comparisons
Lesson 7-8 Prediction Equations
407
CH
APTER
7
Study Guide and Review
wnload Vocabulary view from pre-alg.com
Key Vocabulary X
Be sure the following Key Concepts are noted in your Foldable.
Key Concepts Functions
/
s
W
(Lesson 7-1)
• In a function, each member in the domain is paired with exactly one member in the range.
Representing Linear Functions
(Lesson 7-2)
• A solution of a linear equation is an ordered pair that makes the equation true. • A linear equation can be represented by a set of ordered pairs, a table of values, or a graph.
Rate of Change and Slope
(Lessons 7-3, 7-4, and 7-5)
• A change in one quantity in relation to another quantity is called the rate of change. • When a quantity increases over time, it has a positive rate of change. When a quantity decreases over time, it has a negative rate of change. When a quantity does not change over time, it has a zero rate of change. • Linear relationships have constant rates of change. • Two quantities a and b have a proportional
change in b a linear relationship if _ is constant and _
is constant.
b
constant of variation (p. 378) constant rate of change (p. 376) direct variation (p. 378) family of functions (p. 395) function (p. 359) line of fit (p. 403) linear equation (p. 365) linear relationship (p. 376) rate of change (p. 371) slope (p. 384) slope-intercept form (p. 391) vertical line test (p. 360) y-intercept (p. 391)
change in a
• Slope can be used to describe rates of change. • Slope is the ratio of the rise, or the vertical change, to the run, or the horizontal change.
Writing and Predicting Linear Equations (Lessons 7-6, 7-7, and 7-8)
Vocabular y Check Choose the term that best matches each statement or phrase. Choose from the list above. 1. a relation in which each member of the domain is paired with exactly one member of the range 2. a value that describes the steepness of a line 3. can be drawn through data points to approximate a linear relationship 4. a description of how one quantity changes in relation to another quantity 5. a graph of this is a straight line 6. a linear equation that describes rate of change
• In the slope-intercept form y = mx + b, m is the slope and b is the y-intercept.
7. a way to determine whether a relation is a function
• You can write a linear equation by using the slope and y-intercept, two points on a line, a graph, a table, or a verbal description.
8. the rate of change between any two data points is the same
• A line of fit is used to approximate data.
408 Chapter 7 Functions and Graphing
9. k in the equation y = kx 10. an equation written in the form y = mx + b
Vocabulary Review at pre-alg.com
Lesson-by-Lesson Review 7–1
Functions
(pp. 359–363)
Determine whether each relation is a function. Explain. 11. {(1, 12), (-4, 3), (6, 36), (10, 6)}
Example 1 Determine whether {(-9, 2), (1, 5), (1, 10)} is a function. Explain. $OMAIN X
2ANGE Y
£
Ó x £ä
12. {(11.8, -9), (10.4, -2), (11.8, 3.8)} 13. {(0, 0), (2, 2), (3, 3), (4, 4)} 14. {(-0.5, 1.2), (3, 1.2), (2, 36)} GASOLINE Use the table that shows the cost of gas in different years.
Year 2002 2003 2004
Cost $1.36 $1.59 $1.82
Source: The World Almanac
15. Is the relation a function? Explain. 16. Describe how the cost of gas is related to the year.
This relation is not a function because 1 in the domain is paired with two range values, 5 and 10. Example 2 The table shows the number of Alternative Fuel Vehicles (AFVs). Do these data represent a function? Explain.
Year 2001 2002 2003
AFVs 623,043 895,984 930,538
Source: eia.doe.gov
This relation is a function because during each year, there is only one value of AFVs.
7–2
Linear Equations in Two Variables
(pp. 365–369)
Graph each equation by plotting ordered pairs.
Example 3 Graph y = -x + 2 by plotting ordered pairs.
17. y = x + 4
18. y = x - 2
19. y = -x
20. y = 2x
The ordered pair solutions are: (0, 2), (1, 1), (2, 0), (3, -1).
21. CANDY A regular fruit smoothie x costs $1.50, and a large fruit smoothie y costs $3. Find two solutions of 1.5x + 3y = 12 to determine how many of each type of fruit smoothie Lisa can buy with $12.
Then plot and connect the points. y (0, 2) (1, 1) y x 2 (2, 0) x O (3, 1)
Chapter 7 Study Guide and Review
409
CH
A PT ER
7 7–3
Study Guide and Review
Rate of Change
(pp. 371–375)
Find the rate of change for each function. 22. Time (s) Distance (m) x 0 1 2
y 0 8 16
7–4
5
4
4
8
3
change in time
0 min - 4 min 1 ft or _ 1 ft/min = -4 min -4 1 The rate of change is _ ft/min, or a -4 1 decrease of foot per minute. 4
Constant Rate of Change and Direct Variation Time (min) x 0 20 40 60
Distance (mi) y 0 3 6 9
25. FRUIT The cost of peaches varies directly with the number of pounds bought. If 3 pounds of peaches cost $4.50, find the cost of 5.5 pounds.
Slope
5 ft - 4 ft = __
y 45 43 41
24. Find the constant rate of change for the linear function and interpret its meaning.
7–5
Time Water (min) Level (ft)
change in water level rate of change = __
23. Time (h) Temperature (° F) x 1 2 3
Example 4 The table shows the relationship between time and water level of a pool. Find the rate of change.
(pp. 376–381)
Example 5 Find the rate of change in population from 2000 to 2004 for El Paso, Texas. y2 - y1 rate of change = _ x -x 2
x
Population (1000s) y
2000 2004
564 592
Year
1
Definition of slope
592 - 564 Substitute. =_ 2004 - 2000
=7 Simplify. So, the rate of change in population was 7 thousand people per year.
(pp. 384–389)
Find the slope of the line that passes through each pair of points.
Example 6 Find the slope of the line that passes through A(0, 6) and B(4, -2).
26. J(3, 4), K(4, 5)
y2 - y1 m=_ x -x
Definition of slope
-2 - 6 m=_
(x1, y1) = (0, 6) (x2, y2) = (4, -2)
-8 or -2 m=_
The slope is -2.
27. C(2, 8), D(6, 7)
2
28. ANIMALS A lizard is crawling up a hill that rises 5 feet for every horizontal change of 30 feet. Find the slope.
410 Chapter 7 Functions and Graphing
1
4-0 4
Mixed Problem Solving
For mixed problem-solving practice, see page 800.
7–6
Slope-Intercept Form
(pp. 391–394)
Graph each equation using the slope and y-intercept. 29. y = -x + 4 30. y = -2x + 1
Example 7 State the slope and yintercept of the graph of y = -2x + 3.
31. y = 1x - 2 3
Writing Linear Equations
y = mx + b
33. BIRDS The altitude in feet y of an albatross who is slowly landing can be given by y = 400 - 100x, where x represents the time in minutes. State the slope and y-intercept of the graph of the equation and describe what they represent.
7–7
32. x + y = -5
y = -2x + 3
The slope of the graph is -2, and the y-intercept is 3.
(pp. 397–402)
Write an equation in slope-intercept form for each line. 34. slope = -1, y-intercept = 3 35. slope = 6, y-intercept = -3 Write an equation in slope-intercept form for the line passing through each pair of points. 36. (3, 7), (4, 4) 37. (1, 5), (2, 8) BIRTHDAYS For Exercises 38 and 39, use the following information. It costs $100 plus $30 per hour to rent a movie theater for a birthday party. 38. Write an equation in slope-intercept form that shows the cost y for renting the theater for x hours. 39. Find the cost of renting the theater for 4 hours.
Example 8 Write an equation in slope-intercept form for the line that passes through (5, 9) and (2, 0). Step 1 Find the slope m. y2 - y1 m= _ x -x 2
1
9-0 m=_ or 3 5-2
Step 2
Find the y-intercept b. Use the slope and the coordinates of either point. y = mx + b 9 = 3(5) + b -6 = b
Step 3
Definition of slope (x1, y1) = (5, 9) (x2, y2) = (2, 0)
Slope-intercept form Substitute. Simplify.
Substitute the slope and y-intercept. y = mx + b y = 3x + (-6) y = 3x - 6
Slope-intercept form Substitute. Simplify.
Chapter 7 Study Guide and Review
411
CH
A PT ER
7 7–8
Study Guide and Review
Prediction Equations
(pp. 403–407)
ART The table shows the attendance for an annual art festival. Year 2002 2003 2004 2005
Attendance 2500 2650 2910 3050
Example 9 Make a scatter plot and draw a line of fit for the table showing the attendance at home games for the first four games of a high school football season. Game 1 2 3 4
40. Make a scatter plot and draw a line of fit.
HOUSING The table shows the changes in the median price of existing homes. Year
Median Price ($ thousands)
1991
97.1
1995
110.5
1998
128.4
2000
139.0
2002
158.1
2003
170.0
Draw a line that fits the data. ÌÌi`>Vi Õ`Ài`î
41. Use the line of fit to predict art festival attendance in 2010.
£È £x £{ £Î £Ó ££ ä
Source: National Association of REALTORS
42. Make a scatter plot and draw a line of fit for the data. 43. Use the line of fit to predict the median price for an existing home for the year 2015.
Attendance 1100 1200 1300 1500
£
Ó Î { >i
x
Use the line of fit to predict the attendance for the seventh home game. Extend the line so that you can find the y-value for an x-value of 7. The y-value for 7 is about 19. So, a prediction for the attendance at the seventh home game is approximately 1900 people. Óä
44. Use the line of fit to predict the value of y when x = 7. y
ÌÌi`>Vi Õ`Ài`î
£ £n £Ç £È £x £{ £Î £Ó ££ O
x ä
412 Chapter 7 Functions and Graphing
£
Ó
Î
{ x >i
È
Ç
n
CH
A PT ER
7
Practice Test
Determine whether each relation is a function. Explain.
1. {(-3, 4), (2, 9), (4, -1), (-3, 6)} 2. {(1, 2), (4, -6), (-3, 5), (6, 2)} 3. {(7, 0), (9, 3), (11, 1), (13, 0)}
F y = 3x + 10
Graph each equation by plotting ordered pairs.
4. y = 2x + 1
5. 3x + y = 4
6. MULTIPLE CHOICE Find the rate of change for the linear function represented in the table. Hours Worked Money Earned ($)
1 5.50
2 11.00
3 16.50
16. MULTIPLE CHOICE Victor works at a barber shop. He gets paid $10 an hour plus $3 for every hair cut he performs. Which equation represents Victor’s hourly earnings?
4 22.00
A increase $6.50/h
G y = 10x + 3 H y = 3x - 10 J
y = 10x - 3
RECYCLING For Exercises 17 and 18 use the graph and the information below. Ramiro collected 150 pounds of cans to recycle. He plans to collect an additional 30 pounds each week. The graph shows the amount of cans he plans to collect. {ää
B increase $5.50/h
Îxä Îää
D decrease $6.50/h
Óxä
7. JOBS Determine whether a proportional linear relationship exists between the two quantities in Exercises 6. Explain your reasoning.
>Ã L®
C decrease $5.50/h
£xä £ää
Find the slope of the line that passes through each pair of points. 10. A(2, 5), B(4, 11) 12. F(8, 5), G(7, 9)
11. C(-4, 5), D(6, -3) 13. H(11, 6), J(9, -1)
State the slope and y-intercept of the graph of each equation. Then graph each equation using the slope and y-intercept. 2 x-4 14. y = _ 3
15. 2x + 4y = 12
Chapter Test at pre-alg.com
ä] £xä®
xä ä
FUND-RAISING The total profit for a school varies directly with the number of potted plants sold. Suppose the school earns $57.60 if 12 plants are sold. 8. Write an equation that could be used to find the profit per plant sold. 9. Find the total profit if 65 plants are sold.
Î] Ó{ä®
Óää
£
Ó Î { ÕLiÀ v 7iiÃ
x
17. Find the equation of the line. 18. What does the slope of the line represent? GARDENING For Exercises 19 and 20, use the table and the information below. The full-grown height of a tomato plant and the number of tomatoes it bears are recorded for five tomato plants. 19. Make a scatter plot of the Height Number of data and draw a line of fit. (in.) Tomatoes 27 12 20. Use the line of fit to predict 33 18 the number of tomatoes a 43-inch tomato plant will 19 9 bear. 40 16 31
Chapter 7 Practice Test
15
413
CH
A PT ER
Standardized Test Practice
7
Cumulative, Chapters 1–7
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The graph of the line y = 2x + 2 is shown on the coordinate grid below. n Ç È x { Î Ó £ n ÇÈ x{ÎÓ£" £ Ó Î { x È Ç n
Y
Question 3 When answering GRIDDABLE questions, first fill in the answer on the top row. Then pencil in exactly one bubble under each number or symbol.
£ Ó Î { x È Ç nX
Which table of ordered pairs contains only points on this line? A x C y x y
B
2
2
2
3
1
1
1
2
1
1
4
1
3
x
y
1
3. GRIDDABLE A bus traveled 209 miles at an average speed of 70 miles per hour. About how many hours did it take for the bus to reach its destination? Round your answer to the nearest half hour.
D
x
y
1
1
3
1
1
6
1
3
2
9
2
5
2. Mario has 125 coins in his collection. He plans to add another 5 coins each week until he has doubled the amount in his collection. Which equation can be used to determine w, the number of weeks it will take to double the size of the coin collection? H 5w + 125w = 250 F 5w + 125 = 125 G 5w + 125 = 250 J 2(5w + 125) = 250 414 Chapter 7 Functions and Graphing
4. Chris, Candace, Jamil, and Lydia ate breakfast at a restaurant. The total amount of the bill, including tax and tip was $46.60. 1 of the bill, Chris paid $10, Candace paid _ 4 Jamil paid 20% of the bill, and Lydia paid the rest. Who paid the greatest amount? A Chris B Candace C Jamil D Lydia 5. Mr. Williams wants to purchase some blank CDs. He compared the prices from four different office supplies stores. Which store’s prices are based on a constant unit price? H Number Total F Number Total of CDs 10
Cost $3.65
of CDs 10
Cost $2.50
20
$6.65
20
$5.00
30
$9.65
30
$7.50
$12.65
40
$10.00
Number of CDs 10
Total Cost $3.00
40
G Number
J
of CDs 10
Total Cost $3.50
20
$7.00
20
$5.50
30
$10.00
30
$7.50
40
$12.00
40
$10.00
Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
6. The Saturn V rocket that took Apollo astronauts to the moon weighed 6,526,000 pounds at lift-off. Write its weight in scientific notation. A 6.526 × 10-7 C 6.526 × 106 B 6.526 × 10-6 D 6.526 × 107
9. Which of the following statements is true? F 0.4 > 40% H 40% > 0.04 G 0.04 = 40% J 40% ≤ 0.04 10. The cost, c, of hiring a plumber can be found using the equation c = 75 + 40h, where h is the number of hours the plumber worked. For how many hours was the plumber hired if he charged $215? C 3.75 h A 3h B 3.5 h D 4.25 h
7. Mary-Ann saved $56 when she purchased a television on clearance at an electronics store. If the sale price was 20% off the regular price, what was the regular price? F $250 H $275 G $260 J $280
11. The expression 2(n + 4) describes a pattern of numbers. What is the tenth term of the sequence? F 2 H 25 G 15 J 28
8. The graph shows the shipping charges per order, based on the number of items shipped in the order. Which statement best describes this graph?
Pre-AP Record your answers on a sheet of paper. Show your work.
Shipping Charge ($)
y 18 16 14 12 10 8 6 4 2 0
12. Krishnan is considering three plans for cellular phone service. The plans each offer the same services for different monthly fees and different costs per minute. Plan
Monthly Fee
X
$0
x
1 2 3 4 Number of Items
A As the number of items increases, the shipping charge decreases.
Cost per Minute $0.24
Y
$15.95
$0.08
Z
$25.95
$0.04
a. For each plan, write an equation that shows the total monthly cost c for m minutes of calls. b. What is the cost of each plan if Krishnan uses 100 minutes per month? c. Which plan costs the least if Krishnan uses 100 minutes per month? d. Which plan costs the least if Krishnan uses 300 minutes per month?
B As the number of items increases, the shipping charge increases. C As the number of items decreases, the shipping charge increases. D There is no relationship between the number of items shipped and the shipping charge.
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
Go to Lesson...
7-2
7-7
3-8
5-2
6-1
4-7
6-8
7-1
6-5
1-5
3-7
7-6
Chapters 7 Standardized Test Practice
415
Equations and Inequalities
8 •
Select and use appropriate operations to solve problems and justify solutions.
•
Make connections among various representations of a numerical relationship.
•
Use graphs, tables, and algebraic representations to make predictions and solve problems.
Key Vocabulary null or empty set (p. 426) identity (p. 426) inequality (p. 430)
Real-World Link Capacity You can use an inequality to express the maximum number of people that can be held in the Radio City Music Hall in New York City.
Equations and Inequalities Make this Foldable to help you organize notes on equations and 1 inequalities. Begin with a plain sheet of 8_” by 11” paper. 2
1 Fold in half lengthwise.
2 Fold in thirds and then fold each third in half.
3 Open. Cut one side
4 Label each tab with a
along the folds to make tabs.
lesson number as shown.
416 Chapter 8 Equations and Inequalities CORBIS SYGMA
GET READY for Chapter 8 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Solve each equation. Check your solution.
Example 1
(Lesson 3-3)
Solve _r + 6 = 5.
1. 2x + 5 = 13
2. 4n - 3 = 5
d 3. 16 = 8 + _
c 4. _ + 3 = -9
5. -18 = 4b + 10
h 6. -9 - _ =5
3
-4
4
7. SALES Suppose a computer costs $600. If Tony pays a down payment of $150 and plans to pay the balance in 6 equal installments, how much would each installment be? (Lesson 3-3)
Find each sum or difference. (Lessons 2-2 and 2-3)
8. -28 + (-16)
4
_r + 6 = 5 4
_r + 6 - 6 = 5 - 6 4
_r = -1
4 r 4 _ = 4(-1) 4
r = -4
Write the equation. Subtract 6 from each side. Simplify. Undo division. Multiply each side by 4. Simplify.
Example 2
Find -30 - (-42). 9. 17 + (-25)
10. -13 + 24
11. 36 + (-18)
12. 31 - 48
13. -16 - 7
14. 4 - (-12)
15. -23 - (-29)
-30 - (-42) = -30 + 42 To subtract -42, add 42.
= 12
Simplify.
16. STOCKS The stock market fell 507.99 points on October 19, 1987. If the stock market began the day at a 2246.73, what was its value at the end of the day? (Lesson 2-3)
Find each product or quotient. (Lessons 2-4 and 2-5)
Example 3
Find 6 × (-15).
17. -6(8)
18. -3 · 5
19. -6(-25)
20. 2(-4)(-9)
21. 64 ÷ (-32)
22. -15 ÷ 3
23. -12 ÷ (-3)
24. 24 ÷ (-2)
6 × (-15) = -90 The factors have different signs so the product is negative.
25. CHEMISTRY A solution cooled at a rate of 6˚F every 5 minutes. What was the change 1 hour? (Lesson 2-5) in temperature after _ 2
Chapter 8 Get Ready for Chapter 8
417
EXPLORE
8-1
Algebra Lab
Equations with Variables on Each Side In Chapter 3, you used algebra tiles and an equation mat to solve equations in which the variable was on only one side of the equation. You can use algebra tiles and an equation mat to solve equations with variables on each side of the equation.
ACTIVITY 1 The following example shows how to solve x + 3 = 2x + 1 using algebra tiles. Step 1
Model the equation. 1
x
1
x
x 1
1 x3
Step 2
Remove the same number of x-tiles from each side of the mat until there is an x-tile by itself on one side.
2x 1
1
x
1
x
x 1
1 x x3
2x x 1
Variables Keep in mind that, unlike the numerical tiles, the size of the x-tile does not correspond to the number that it represents. Remember that the x-tile represents an unknown number, and so even though the x-tile appears to be only slightly larger than the 1-tile, x could actually be much larger, or even smaller, than 1.
Step 3
Remove the same number of positive tiles from each side of the mat until the x-tile is by itself on one side.
1
1
x 1
1
2
x
There are two positive tiles on the left side of the mat and one x-tile on the right side. Therefore, x = 2. Since 2 + 3 = 2(2) + 1, the solution is correct.
ANALYZE THE RESULTS Use algebra tiles to model and solve each equation. 1. 2x + 3 = x + 5
2. 3x + 4 = 2x + 8
3. 3x = x + 6
4. 6 + x = 4x
5. 2x - 4 = x - 6
6. 5x - 1 = 4x - 5
7. Which property of equality allows you to remove a positive tile from each side of the mat? 8. Explain why you can remove an x-tile from each side of the mat. 418 Chapter 8 Equations and Inequalities
Some equations are solved by using zero pairs. Remember, you may add or subtract a zero pair from either side of an equation mat without changing its value. The following example shows how to solve 2x + 1 = x - 5.
Review Vocabulary Zero Pair A pair of numbers that, when added together, equal zero. Example: 3 + -3 = 0
ACTIVITY 2 Step 1 Model the equation.
1 x
x
x
1 2x 1
Step 2
Remove the same number of x-tiles from each side of the mat until there is an x-tile by itself on one side.
Step 4
Remove the zero pair from the left side. There are 6 negative tiles on the right side of the mat.
1
1
1 x
x
2x x 1
It is not possible to remove the same number of 1-tiles from each side of the mat. Add 1 negative tile to the left side to make a zero pair. Add 1 negative tile to the right side of the mat.
1
x 5
x
1
Step 3
1
1
1
1
1
x x 5
1 1 1
x
1 1
1
x 1 (1)
1 x
1 1
5 (1)
1 1
1
x
1 1 1 1
6
Therefore, x = -6. Since 2(-6) + 1 = -6 - 5, the solution is correct.
ANALYZE THE RESULTS Use algebra tiles to model and solve each equation. 9. 2x + 3 = x - 5
10. 3x - 2 = x + 6
11. x - 1 = 3x + 7
12. x + 6 = 2x - 3
13. 2x + 4 = 3x - 2
14. 4x - 1 = 2x + 5
15. Does it matter whether you remove x-tiles or 1-tiles first? Explain. 16. Explain how you could use models to solve -2x + 5 = -x - 2. Explore 8-1 Algebra Lab: Equations with Variables on Each Side
419
8-1
Solving Equations with Variables on Each Side
Main Idea • Solve equations with variables on each side.
Each bag on the balance contains the same number of blocks. (Assume that the paper bag weighs nothing.) a. The two sides balance. Without looking in a bag, how can you determine the number of blocks in each bag? b. Explain why your method works. c. Suppose x represents the number of blocks in the bag. Write an equation that is modeled by the balance. d. Explain how you could solve the equation.
Equations with Variables on Each Side To solve equations with Look Back To review Addition and Subtraction Properties of Equality, see Lesson 3-3.
variables on each side, use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side. Then solve the equation.
EXAMPLE
Equations with Variables on Each Side
Solve 2x + 3 = 3x. Check your solution. 2x + 3 = 3x 2x - 2x + 3 = 3x - 2x 3=x Subtract 2x from the left side of the equation to isolate the variable.
Write the equation. Subtract 2x from each side. Simplify.
Subtract 2x from the right side of the equation to keep it balanced.
To check your solution, replace x with 3 in the original equation. CHECK
2x + 3 = 3 x 2(3) + 3 3(3) 6+39 9=9
Write the equation. Replace x with 3. Simplify. The statement is true.
The solution is 3.
1. Solve 7x = 5x + 4. Check your solution. 420 Chapter 8 Equations and Inequalities
Extra Examples at pre-alg.com
EXAMPLE BrainPOP® at pre-alg.com
Equations with Variables on Each Side
Solve each equation. Check your solution. a. 5x + 4 = 3x - 2 Write the equation. 5x + 4 = 3x - 2 5x - 3x + 4 = 3x - 3x - 2 Subtract 3x from each side. 2x + 4 = -2 Simplify. 2x + 4 - 4 = -2 - 4 Subtract 4 from each side. 2x = -6 Simplify. x = -3 Check your solution.
b. 2.4 + a = 2.5a - 4.5 Write the equation. 2.4 + a = 2.5a - 4.5 2.4 + a - a = 2.5a - a - 4.5 Subtract a from each side. 2.4 = 1.5a - 4.5 Simplify. 2.4 + 4.5 = 1.5a - 4.5 + 4.5 Add 4.5 to each side. 6.9 = 1.5a Simplify.
6.9 1.5a _ =_
Divide each side by 1.5.
4.6 = a
Check your solution.
1.5
1.5
2A. 2x + 3 = 3x - 2
2B. 3.2 + 0.3x = 0.2x + 1.4
RENTALS Under Plan A, an annual membership costs $30 plus $1.50 for each DVD rental. Under Plan B, the annual membership costs $12 plus $3 for each DVD rental. What number of DVD rentals results in the same yearly cost? Let v represent the number of videos rented. $30 plus $1.50 for each video Real-World Link In 1980, only 1% of American households owned a VCR. Today more than 86% own either a VCR or DVD player. Source: Statistical Abstract of the United States
equals
$12 plus $3 for each video
30 + 1.50v = 12 + 3v 30 + 1.50v = 12 + 3v Write an equation. 30 + 1.5v - 1.5v = 12 + 3v - 1.5v Subtract 1.5v from each side. 30 = 12 + 1.5v Simplify. 30 - 12 = 12 - 12 + 1.5v Subtract 12 from each side. 18 = 1.5v Simplify. 18 1.5v _ =_ 1.5
1.5
12 = v
Divide each side by 1.5. Simplify.
The yearly cost is the same for 12 rentals.
3. CRUISES Red Bird Cruises charges $85 per day plus a one-time fee of $75 for taxes and gratuities. King Cruises charges $100 per day plus a fee of $30. For what number of days do the cruise companies charge the same? Personal Tutor at pre-alg.com Lesson 8-1 Solving Equations with Variables on Each Side Jose Luis Pelaez, Inc./CORBIS
421
Solve each equation. Check your solution. Example 1 (p. 420)
Example 2 (p. 421)
Example 3 (p. 421)
HOMEWORK
HELP
For See Exercises Examples 10–13 1 14–19 2 20–23 3
1. 4x - 8 = 5x
2. 4x + 9 = 7x
3. 12x = 2x + 40
4. 6a = 26 + 4a
5. 4x - 1 = 3x + 2
6. 4k + 24 = 6k - 10
7. 7.2 - 3c = 2c - 2
8. 3 - 3.7b = 10.3b + 10
9. CAR RENTAL Suppose you can rent a car from ABC Auto for either $25 a day plus $0.45 a mile or for $40 a day plus $0.25 a mile. What number of miles results in the same cost for one day?
Solve each equation. Check your solution. 10. 2x + 3 = x
11. n - 14 = 3n
12. 8 - 2c = 2c
13. q - 2 = -q + 1
14. 13y - 18 = -5y + 36
15. -s + 4 = 7s - 3
16. 7d - 13 = 3d + 7
17. 2f - 6 = 7f + 24
18. 12n - 23.2 = -14n + 28.8
19. 3.1w + 5 = 0.8 + w
Define a variable and write an equation to find each number. Then solve. 20. Twice a number is 220 less than six times the number. What is the number? 21. Fourteen less than three times a number equals the number. Find the number. 22. GEOGRAPHY South Carolina’s coastline is 358 kilometers longer than twice the coastline of North Carolina. It is also 842 kilometers longer than the coastline of North Carolina. Find the lengths of the coastlines of South Carolina and North Carolina. 23. MUSIC DOWNLOADS Denzel is comparing Web sites for downloading music. One charges a $5 membership fee plus $0.50 per track. Another charges $1.00 per track, but has no monthly fee. How many songs would Denzel have to buy for him to spend the same amount at both Web sites? Solve each equation. Check your solution.
EXTRA
PRACTICE
See pages 779, 801. Self-Check Quiz at pre-alg.com
H.O.T. Problems
24. 12 + 1.5a = 3a
25. 12.6 - x = 2x
26. 2b + 6.2 = 13.2 - 8b
27. 3c + 4.5 = 7.2 - 6c
28. 12.4y + 14 = 6y - 2
29. 4.3n - 1.6 = 2.3n + 5.2
30. 0.4x = 2x + 1.2
1 1 b + 8 =_ b-4 31. _ 3
2
32. CELLULAR PHONES One cellular phone carrier charges $29.75 a month plus $0.15 a minute for international calls. Another carrier charges $19.95 a month and $0.29 a minute for international calls. For how many minutes is the cost of the plans the same? 33. NUMBER SENSE Three times the quantity y + 7 is equal to four times the quantity y - 2. What value of y makes the sentence true?
422 Chapter 8 Equations and Inequalities
34. OPEN ENDED Write an example of an equation with variables on each side. State the steps you would use to isolate the variable.
The trends in attendance at various sporting events can be represented by equations. Visit pre-alg.com.
35. CHALLENGE An empty bucket is put under two faucets. If one faucet is turned on alone, the bucket fills in 6 minutes. If the other faucet is turned on alone, the bucket fills in 4 minutes. If both are turned on, how many seconds will it take to fill the bucket? 36.
Writing in Math
Explain how solving equations with variables on each side is like solving equations with variables on just one side. Include examples of both types of equations and an explanation of how they are alike and how they are different.
9 37. The formula F = _ C + 32 is used to 5 find the Fahrenheit temperature when a Celsius temperature is known. For what value are the Celsius and Fahrenheit temperatures the same?
A -72°
39. Olivia’s manager gave her a choice as to how she wants to be paid.
C 0°
B -40° D 32° 38. Two weeks ago the sewing club had 1 less than 3 times their average attendance. Last week they had 3 more than their average attendance. If the attendance for both weeks were equal, what is the average attendance of the sewing club? F 1
H 3
G 2
J 4
Pay per Hour
Pay for Each Dollar of Appliance Sales
Plan 1
$3
15¢
Plan 2
$4
10¢
Which equation shows what Olivia’s sales need to be in one hour to earn the same amount under either plan? A 3 + 0.15s = 4 + 0.10s B 3s + 0.15 = 4s + 0.10 C 3 + 0.10s = 4 + 0.15s D 3(s + 0.15) = 4(s + 0.10)
40. Find the true statement. (Lesson 7-8) • A line of fit is close to most of the data points. • A line of fit describes the exact coordinates of each point in the data set. • A line of fit always has a positive slope. 41. What equation represents the table of values? (Lesson 7-7)
x
-4
-8
-12
-16
y
6
8
10
12
PREREQUISITE SKILL Use the Distributive Property to rewrite each expression as an equivalent algebraic expression. (Lesson 3-1) 42. 4(x - 8)
43. 2(1.2c + 14)
1 44. _ (n - 9) 2
Lesson 8-1 Solving Equations with Variables on Each Side
423
8-2
Solving Equations with Grouping Symbols
Main Ideas • Solve equations that involve grouping symbols. • Identify equations that have no solution or an infinite number of solutions.
New Vocabulary null or empty set identity
Josh starts walking toward the park at a rate of 2 mph. One hour later, his sister Maria starts on the same path, riding her bike at 10 mph. The table shows expressions for the distance Maria and Josh have traveled after a given time.
Rate (mph)
a. What does t represent?
Time (hours)
Distance (miles)
Josh
2
t
2t
Maria
10
t-1
10(t - 1)
b. Why is Maria’s time shown as t - 1? c. Write an equation that represents the time when Maria catches up to Josh. (Hint: They will have traveled the same distance.)
Solve Equations with Grouping Symbols To find how many hours it takes Maria to catch up to Josh, you can solve the equation 2t = 10(t - 1). First, use the Distributive Property to remove the grouping symbols.
EXAMPLE
Solve Equations with Parentheses
a. Solve the equation 2t = 10(t - 1). Check your solution. 2t = 10(t - 1)
Write the equation.
2t = 10(t) - 10(1)
Use the Distributive Property.
2t = 10t - 10
Simplify.
2t - 10t = 10t - 10t - 10
Review Vocabulary Dimensional Analysis The process of including units of measurement when computing (Lesson 5-3)
Subtract 10t from each side.
-8t = -10
Simplify.
-10 -8t _ =_
Divide each side by -8.
-8 5 1 t = _ or 1_ 4 4
-8
CHECK
Simplify.
Use dimensional analysis. 2 miles _ 1 · 5 hour or 2 _ miles. Josh traveled _ 2
4
hour
Maria traveled one hour less than Josh. She traveled 10 miles _ 1 _ · 1 hour or 2 _ miles. hour
4
2
1 hour, or 15 minutes. Therefore, Maria caught up to Josh in _ 4
424 Chapter 8 Equations and Inequalities
Extra Examples at pre-alg.com
b. Solve 5(a - 4) = 3(a + 1.5).
Alternative Method You can also solve the equation by subtracting 3a from each side first, then adding 20 to each side.
5(a - 4) = 3(a + 1.5) 5a - 20 = 3a + 4.5 5a - 20 + 20 = 3a + 4.5 + 20 5a = 3a + 24.5 5a - 3a = 3a - 3a + 24.5 2a = 24.5 24.5 2a _ =_
Use the Distributive Property. Add 20 to each side. Simplify. Subtract 3a from each side. Simplify. Divide each side by 2.
2
2
Write the equation.
a = 12.25
Simplify.
Solve each equation. Check your solution. 1A. 3x = 4(x + 2) 1B. -0.2(3c + 15) = 3(0.8c - 8)
Sometimes a geometric figure is described in terms of only one of its dimensions. To find the dimensions, you may have to solve an equation that contains grouping symbols.
EXAMPLE
Use an Equation to Solve a Problem
GEOMETRY The perimeter of a rectangle is 46 inches. Find the dimensions if the length is 5 inches greater than twice the width. Words Variable Equation
Review Vocabulary Perimeter The distance around a geometric figure; Example: The perimeter of a square with sides that are 5 inches long is 20 inches. (Lesson 3-8)
2 times width
+
2 times length
= perimeter w
Let = the width. Let 2w + 5 = the length. 2w
+
2w + 2(2w + 5) = 46 2w + 4w + 10 = 46 6w + 10 = 46 6w + 10 - 10 = 46 - 10 6w = 36 w=6
2(2w + 5)
=
46
2w + 5
Write the equation. Use the Distributive Property. Simplify. Subtract 10 from each side. Simplify. Mentally divide each side by 6.
Evaluate 2w + 5 to find the length. 2(6) + 5 = 12 + 5 or 17
Replace w with 6.
The width is 6 inches. The length is 17 inches.
2. RECYCLING Sofia recycled 3 pounds less than 3 times the amount that James recycled. If they recycled a total of 53 pounds, how many pounds did each person recycle? Personal Tutor at pre-alg.com Lesson 8-2 Solving Equations with Grouping Symbols
425
No Solution or All Numbers as Solutions Some equations have no solution. That is, no value of the variable results in a true sentence. When this occurs, the set of solutions for the equation contains no elements. A set that contains no elements is called the null or empty set, shown by the symbol ∅ or {}.
EXAMPLE
No Solution
1 1 Solve 3x + _ = 3x - _ .
Interactive Lab pre-alg.com
3 2 1 1 Write the equation. 3x + _ = 3x - _ 3 2 1 1 3x - 3x + _ = 3x - 3x - _ Subtract 3x from each side. 3 2 _1 = -_1 Simplify. 3 2 1 1 The sentence _ = -_ is never true. So, the solution set is ∅. 3 2
3. Solve 6x + 4 = 2(3x - 5). Check your solution. An equation that is true for every value of the variable is called an identity.
EXAMPLE
All Numbers as Solutions
Solve 2(2x - 1) + 6 = 4x + 4. 2(2x - 1) + 6 = 4x + 4 Write the equation. 4x - 2 + 6 = 4x + 4 Use the Distributive Property. 4x + 4 = 4x + 4 Simplify. 4x + 4 - 4 = 4x + 4 - 4 Subtract 4 from each side. 4x = 4x Simplify. x=x Mentally divide each side by 4. The sentence x = x is always true. The solution set is all numbers.
4. Solve 20f + (-8f - 15) = 3(4f - 5). Check your solution.
Example 1 (pp. 424–425)
Example 2 (p. 425)
Examples 3, 4 (p. 426)
Solve each equation. Check your solution. 1. 3(g - 3) = 6
2. 4(x + 1) = 28
3. 2(a - 2) = 3(a - 5)
4. 16(z + 3) = 4(z + 9)
5. 5(2c + 7) = 80
6. 6(3d + 5) = 75
7. GEOMETRY The perimeter of a rectangle is 20 feet. The width is 4 feet less than the length. Find the dimensions of the rectangle. Solve each equation. Check your solution. 8. 12 - h = -h + 3 10. 3(2g + 4) = 6(g + 2)
426 Chapter 8 Equations and Inequalities
9. 3n + 4 = 3(n + 2) 11. 4(f + 3) + 5 = 17 + 4f
HOMEWORK
HELP
For See Exercises Examples 12–19 1 20, 21 3 22, 23 4 24, 25 2
Solve each equation. Check your solution. 12. 2(d + 6) = 3d - 1
13. 6n - 18 = 4(n + 2.1)
14. 3(a - 3) = 2(a + 4)
15. 3(s + 22) = 4(s + 12)
16. 4(x - 2) = 3(1.5 + x)
17. 3(a - 1) = 4(a - 1.5)
18. 2(3.5n + 6) = 2.5n - 2
19. 4.2x - 9 = 3(1.2x + 4)
20. 2(x - 5) = 4x - 2(x + 5) + 1
21. (3x + 2) + (-x + 5) = 2x - 7
22. 8y - 5 = 5(y - 1) + 3y
23. 10z + 4 = 2(5z + 8) - 12
24. GEOMETRY The perimeter of a rectangle is 32 feet. Find the dimensions of the rectangle if the length is 4 feet longer than three times the width. Then find the area of the rectangle. 25. BASKETBALL Camilla has three times as many points as Lynn. Lynn has five more points than Kim. Camilla, Lynn, and Kim combined have twice as many points as Jasmine. If Jasmine has 25 points, how many points does each of the other three girls have? Find the dimensions of each rectangle. The perimeter is given. 26. P = 460 ft
27. P = 440 yd
28. P = 11 m w
w
w
2w - 2 3w - 60
w + 30
Solve each equation. Check your solution. 1 29. _ (2n - 5) = 4n - 1
1 30. y - 2 = _ (y + 6)
1 (24b + 60) 31. -3(4b - 10) = _ 2
3 1 32. _ a+4=_ (3a + 16) 4 4
33. 0.4d = 2d + 1.24
a-6 a-2 =_ 34. _
2
3
12
35. GEOMETRY The triangle and the rectangle have the same perimeter. Find the dimensions of each figure. Then find the perimeter of each figure.
4
x -3 x
x +2
x +1
x +1
EXTRA
PRACTICE
See pages 779, 801. Self-Check Quiz at pre-alg.com
H.O.T. Problems
36. DECORATING A gallon of paint covers about 350 square feet. A painter estimates the area to paint by multiplying the combined wall lengths by the height and subtracting 15 square feet for each window or door. Suppose a rectangular room measures 15 feet long by 12 feet wide. The room is 9 feet high and has two windows and two doors. How many gallons of paint are needed to paint the room using two coats of paint? 37. OPEN ENDED Give an example of an equation that has no solution and an equation that is an identity. 38. CHALLENGE An apple costs the same as 2 oranges. Together, an orange and a banana cost 10¢ more than an apple. Two oranges cost 15¢ more than a banana. What is the cost for one of each fruit? Lesson 8-2 Solving Equations with Grouping Symbols
427
39. SELECT A TOOL/TECHNIQUE Jamie has two spools with an equal length of plastic fencing that she is going to use to fence a rectangular and a triangular section of grass. The length of the rectangle will be 40 feet greater than the width, and the length of each side of the triangle will be 45 feet longer than the width of the rectangle. What technique(s) could be used to find the lengths of the sides? Justify your response and use your technique(s) to solve the problem. draw a model
use paper/pencil
use a calculator
40. NUMBER SENSE Three times the sum of three consecutive integers x, x + 1, and x + 2, is 72. What are the integers? 41.
Writing in Math Why is the Distributive Property important for solving equations? Include in your answer a definition of the Distributive Property and a description of its use in solving equations.
42. Wes leaves downtown driving 55 miles per hour. Emma follows 1 hour later, driving 60 miles per hour. Which equation can be used to determine how long it is after Wes leaves that Emma will catch up? A 55x = 60x - 1
C 55x = 60x
B 60x = 55(x - 1)
D 55x = 60(x - 1)
43. Find the value of x so that the polygons have the same perimeter. F 3 G 6 H 8
X { XÓ
X Î
X { X x
J 12
ALGEBRA Solve each equation. Check your solution. (Lesson 8-1) 44. 4x = 2x + 5
45. 3x + 5 = 7 - 2x
46. 1.5x + 9 = 3x - 3
47. HOUSING The table shows the median price of existing homes. Make a scatter plot and draw a line of fit for the data. Use the line of fit to predict the median price for an existing home in 2010. (Lesson 7-8) Year
1991
1995
1998
2000
2002
2003
Median Price (thousands)
97.1
110.5
128.4
139.0
158.1
170.0
Source: National Association of REALTORS
Express each number in scientific notation. (Lesson 4-7)
48. 4,500,000
49. -37,000
50. 0.000498
PREREQUISITE SKILL Evaluate each expression. (Lesson 1-3) 52. 2t + 8, t = -3 53. b + 11, b = -15 428 Chapter 8 Equations and Inequalities
51. -0.00203
54. 4a, a = -6
Meanings of at Most and at Least The phrases at most and at least are used in mathematics. In order to use them correctly, you need to understand their meanings. Phrase
Meaning
Mathematical Symbol
at most
• no more than • less than or equal to
≤
at least
• no less than • greater than or equal to
≥
Here is an example of one common use of each phrase, its meaning, and a mathematical expression for the situation. Verbal Expression You can spend at most $20. Meaning You can spend $20 or any amount less than $20. Mathematical Expression s ≤ 20, where s represents the amount you spend.
Verbal Expression Meaning
A person must be at least 18 to vote. A person who is 18 years old or any age older than 18 may vote. Mathematical Expression a ≥ 18, where a represents age.
Notice that the word or is part of the meaning in each case.
Reading to Learn 1. Write your own rule for remembering the meanings of at most and at least. For each expression, write the meaning. Then write a mathematical expression using ≤ or ≥. 2. You need to earn at least $50 to help pay for a class trip. 3. The sum of two numbers is at most 6. 4. You want to drive at least 250 miles each day. 5. You want to hike 4 hours each day at most. 6. There are no more than 25 apples in the basket. 7. It will take at least 5 hours to finish this project. Reading Math Meanings of at Most and at Least Bob Daemmrich/The Image Works
429
8-3
Inequalities
Main Ideas • Write inequalities. • Graph inequalities.
New Vocabulary
Children under 6 eat free.
Speed Limit
Must be over 40 inches tall to ride.
inequality
35
a. Name three ages of children who can eat free at the restaurant. Does a child who is 6 years old eat free? b. Name three heights of children who can ride the ride at the amusement park. Can a child who is 40 inches tall ride? c. Name three speeds that are legal. Is a driver who is traveling at 35 mph driving at a legal speed?
Write Inequalities A mathematical sentence that contains , ≤, or ≥ is called an inequality.
EXAMPLE
Write Inequalities
Write an inequality for each sentence. a. Your age is less than 6 years. Words Variable Inequality
Your age
is less than
6 years.
Let a represent your age. a
6
35
1A. Your height is greater than or equal to 40 inches. 1B. Your speed is less than or equal to 35 miles per hour. 430 Chapter 8 Equations and Inequalities
Extra Examples at pre-alg.com
The table below shows some common verbal phrases and the corresponding mathematical inequalities.
Reading Math Inequalities Notice that ≤ and ≥ combine the symbol < or > with part of the symbol for equals, =.
Inequalities
• is less than • is fewer than
≤
• is greater than • is more than • exceeds
≥
• is less than or equal to • is no more than • is at most
• is greater than or equal to • is no less than • is at least
NUTRITION A food can be labeled low fat if it has no more than 3 grams of fat per serving. Write an inequality to describe low-fat foods. Words
Grams of fat per serving is no more than 3.
Variable
Let = number of grams of fat per serving.
Inequality
f
≤
3
The inequality is f ≤ 3.
2. DRIVER’S EDUCATION A student must have at least 10 hours of instructorassisted driving time. Write an inequality to describe this situation.
Inequalities with variables are open sentences. When the variable in an open sentence is replaced with a number, the inequality may be true or false.
EXAMPLE
Determine Truth of an Inequality
For the given value, state whether each inequality is true or false. a. s - 7 < 5, s = 14
Reading Math Inequality Symbols ≮ means is not less than.
s-7 12 is false, the equation 12 = 12 is true. Therefore, this sentence is true.
3A. 3 + x ≤ 12, x = 6
3B. y - 7 < 10, y = 17
Personal Tutor at pre-alg.com Lesson 8-3 Inequalities
431
Graph Inequalities Inequalities can be graphed on a number line. The graph helps you visualize the values that make the inequality true.
EXAMPLE
Graph Inequalities
Graph each inequality on a number line. a. x > 4
b. x ≥ 4
Inequalities When inequalities are graphed, an open dot means the number is not included and a closed dot means it is.
2
3
4
5
6
2
The open circle means the number 4 is not included in the graph.
4
5
6
The closed circle means the number 4 is included in the graph.
c. x < 4 2
3
d. x ≤ 4 3
4
5
6
2
4A. x < 5
4B. x ≥ -2
4C. x > 0
EXAMPLE
Write an Inequality
3
4
5
6
4D. x ≤ 2
Write the inequality for the graph. 4
5
6
7
8
9 10 11 12 13 14
An open circle is on 10, so the point 10 is not included in the graph. The arrow points to the right, so the graph includes all numbers greater than 10. The inequality is x > 10.
5A.
5B. x { Î Ó £ ä £ Ó Î { x
Example 1 (p. 430)
Example 2 (p. 431)
Example 3 (p. 431)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
Write an inequality for each sentence. 1. Lacrosse practice will be no more than 45 minutes. 2. Mario is more than 60 inches tall. 3. SOCCER More than 8000 fans attended the Wizards’ opening soccer game at Arrowhead Stadium in Kansas City, Missouri. Write an inequality to describe the attendance. For the given value, state whether the inequality is true or false. 4. n + 4 > 6, n = 12
432 Chapter 8 Equations and Inequalities
5. 34 ≤ 4r, r = 8
Example 4 (p. 432)
Example 5 (p. 432)
Graph each inequality on a number line. 6. n > 3
HELP
For See Exercises Examples 12–15
1
16, 17
2
18–23
3
24–35
4
36–39
5
8. x < 7
9. d ≥ -6
Write the inequality for each graph. 10.
11. 7
HOMEWORK
7. y ≤ 14
8
-22
9 10 11 12 13 14 15
-20
-18
-16
-14
Write an inequality for each sentence. 12. The elevators in an office building have been approved for a maximum load of 3600 pounds. 13. Kyle’s earnings were no more than $60. 14. The race time of 86 minutes was greater than the winner’s time. 15. After a withdrawal, a savings account is now less than $500. ANALYZE TABLES For Exercises 16 and 17 use the table that shows the average amount of time students ages 14 to 18 spend on homework per week. 16. Inali spends at least an hour more than the average Average Hours Group time spent by boys on homework each week. Write per week an inequality for Inali’s homework time. Male 5.4 17. Anna usually spends no more than the average Female 6.8 time spent by girls on homework each week. Write Source: Horatio Alger Association an inequality to represent Anna’s homework time. For the given value, state whether each inequality is true or false. 18. 18 − x > 4, x = 12
19. 14 + n < 23, n = 8
20. 5k > 35, k = 7
14 < 7, c = 2 21. _ c
y 22. _ ≥ 2, y = 9 3
23. 16 ≤ 3d, d = 8
Graph each inequality on a number line. 24. a > 4
25. x > 6
26. d ≤ 5
27. w ≤ 8
28. n < 11
29. x < 5
30. t ≥ 9
31. b ≥ 8
32. x > -4
33. n ≥ -3
34. x ≤ -5
35. x < -2
Write the inequality for each graph. 36.
37. -10 -9 -8 -7 -6 -5 -4 -3 -2
38.
PRACTICE
See pages 780, 801. Self-Check Quiz at pre-alg.com
1
39. -2 -1
EXTRA
-7 -6 -5 -4 -3 -2 -1
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
40. SPORTS There are more than 32,150 high school girls basketball and track programs in the United States. If there are 15,089 girls track programs, write and solve an inequality to determine the number of girls basketball programs. 41. RESEARCH Use the Internet or another source to find the state or national spending limits on certain government branches, organizations, or projects. Write an inequality to express one or more of these limits. Lesson 8-3 Inequalities
433
H.O.T. Problems
42. NUMBER SENSE Provide a counterexample to the statement, “All numbers less than 0 are negative integers.” 43. OPEN ENDED Write four examples of inequalities, each using one of the symbols , ≤, and ≥. Tell the meaning of each inequality. 44. CHALLENGE Graph the solutions for each compound inequality. a. y < -2 or y > 3 (Hint: In a sentence, or means either part is true.) b. y ≥ 0 and y ≤ 5 (Hint: In a sentence, and means both parts must be true.) 45.
Writing in Math How can inequalities help you describe relationships? Illustrate your answer with a real-world example that uses an inequality symbol and an explanation of the relationships described by the inequality.
46. Young adults are not allowed to vote in elections before their 18th birthday. Which graph represents the age of people who are allowed to vote? A £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ
B £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ
C £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ
47. Which inequality represents the graph below? £ä n Ç È x {
F x ≥ -8 G x ≤ -8 H x > -8 J x < -8
D £Ó £Î £{ £x £È £Ç £n £ Óä Ó£ ÓÓ
ALGEBRA Solve each equation. Check your solution. (Lesson 8-2) 48. 2(3 + x) = 14
49. 63 = 9(2y – 3)
50. 3(n – 1) = 1.5(n + 2)
51. ALGEBRA Four times a number minus 6 is equal to the sum of 3 times the number and 2. Define a variable and write an equation to find the number. (Lesson 8-1) Find each quotient. Write in simplest form. (Lesson 5-4) 3 1 1 7 1 2 52. _ ÷_ 53. _ ÷ _ 54. _ ÷_ 3 3 5 2 2 4 7 1 _ 55. MOWING Keri has gallon of gasoline left. Her mower uses _ gallon to cut 8 6 an average yard. How many average yards can she mow? (Lesson 5-4)
PREREQUISITE SKILL Solve each equation. (Lesson 3-3) 56. x + 19 = 32
57. a + 7 = -3
58. 26 + c = 19
59. 44 – c = 26
60. y – 9.7 = 10.1
61. r – 1.6 = -0.6
434 Chapter 8 Equations and Inequalities
8-4
Solving Inequalities by Adding or Subtracting
Main Idea • Solve inequalities by using the Addition and Subtraction Properties of Inequality.
The paper bag on the balance may contain some blocks. The scale models an inequality because the two sides are not equal. The side with the bag and 2 blocks weighs less than the side with 5 blocks. So, the inequality is x + 2 < 5.
x+2 b, then a + c > b + c and a - c > b - c. 2. if a < b, then a + c < b + c and a - c < b - c.
Inequalities When you add or subtract any number from each side of an inequality, the inequality symbol remains the same.
Examples
23
2+33-4
5 -1
These properties are also true for a ≥ b and a ≤ b.
EXAMPLE
Solve an Inequality Using Subtraction
Solve x + 3 > 10. Check your solution. x + 3 > 10
Write the inequality.
x + 3 - 3 > 10 - 3 Subtract 3 from each side. x>7
Simplify.
(continued on the next page)
Lesson 8-4 Solving Inequalities by Adding or Subtracting
435
To check your solution, try any number greater than 7. CHECK x + 3 > 10
8 + 3 10
Checking Solutions Try a number less than 7 to show that it is not a solution.
Write the inequality. Replace x with 8.
11 > 10 This statement is true. Any number greater than 7 will make the statement true. Therefore, the solution is x > 7.
1. Solve z + 4 > 3. Check your solution.
EXAMPLE
Solve an Inequality Using Addition
Solve -6 ≥ n - 5. Check your solution. -6 ≥ n - 5
Write the inequality.
-6 + 5 ≥ n - 5 + 5 Add 5 to each side. -1 ≥ n
Simplify.
The solution is -1 ≥ n or n ≤ -1.
2. Solve -3 ≥ g - 7. Check your solution.
EXAMPLE
Graph Solutions of Inequalities
1 Solve a + _ < 2. Graph the solution on a number line. 2
1 a+_ 4 9. x + 3.75 ≤ 5
Example 4 (p. 437)
10. 7 > z + 2 3
11. SAVINGS Chris is saving money to buy a stereo. He has $62.50, but his goal is to save at least $100. What is the least amount Chris still needs to save to reach his goal?
Extra Examples at pre-alg.com Courtesy Ohio Expo Center
8. x - 6 ≤ 4
Lesson 8-4 Solving Inequalities by Adding or Subtracting
437
HOMEWORK
HELP
For See Exercises Examples 12–23 1, 2 24–32 3 33, 34 4
Solve each inequality. Check your answer. 12. p + 7 < 9
13. t + 6 > -3
14. -13 ≥ 9 + b
15. 16 > -11 + k
16. 3 ≥ -2 + y
17. 25 < n + (-12)
18. r - 5 ≤ 2
19. a - 6 < 13
20. j - 8 ≤ -12
21. -8 > h - 1
22. 22 > w - (-16)
23. -30 ≤ d + (-5)
Solve each inequality. Then graph the solution on a number line. 24. n + 4 < 9
25. t + 7 > 12
26. p + (-5) > -3
27. -3 + z > 2
28. -13 ≥ x - 8
29. -32 ≥ a + (-5)
1 30. 3 ≤ _ +a
2 31. 4 ≥ s - _
3 32. -_ <w-1
2
3
4
33. TRANSPORTATION A certain minivan has a maximum carrying capacity of 1100 pounds. If the luggage weighs 120 pounds, what is the maximum weight allowable for passengers? 34. MARINE BIOLOGY Manatees can weigh up to 1000 pounds and are generally no more than 10 feet long. Suppose a manatee is currently 6.25 feet long. Write and solve an inequality to find how much longer the manatee could grow. HURRICANES For Exercises 35–37, use the diagram below. Types of Storms Tropical Storm
Depression 39
Hurricane 74
Wind Speed of Storm (mph)
35. A hurricane has winds that are at least 74 miles per hour. Suppose a tropical storm has winds that are 42 miles per hour. Write and solve an inequality to find how much the winds must increase so the storm is a hurricane. 36. Tropical storm Alpha has winds of 50 miles per hour. Write and solve an inequality to find how much the winds need to decrease so that the storm is downgraded to a depression. 37. A major storm has wind speeds that are at least 110 miles per hour. Write and solve an inequality that describes how much greater these wind speeds are than a hurricane with the slowest winds. Solve each inequality. Check your solution. EXTRA
PRACTIICE
38. 1 + y ≤ 2.4
39. 2.9 < c + 7
40. f - 4 ≥ 1.4
41. z - 2 > -3.8
3 1 < 2_ 42. b - _ 2 4
2 1 43. g - 1_ > 2_
See pages 780, 801. Self-Check Quiz at pre-alg.com
3
6
44. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would need to solve an inequality using addition or subtraction.
438 Chapter 8 Equations and Inequalities
H.O.T. Problems
45. OPEN ENDED Write an inequality for the solution graphed below. 16
18
20
22
24
46. FIND THE ERROR Dylan and Jada are using the statement x minus three is greater than or equal to 15 to find values of x. Who is correct? Explain. Jada x - 3 = 15 x - 3 + 3 = 15 + 3 x = 18
Dylan x - 3 ≥ 15 x - 3 + 3 ≥ 15 + 3 x ≥ 18
47. CHALLENGE Is it always, sometimes, or never true that x - 1 < x? Explain. 48.
Writing in Math
How is solving an inequality similar to solving an
equation?
49. Trevor has $25 to spend on a T-shirt and shorts for gym class. The shorts cost $14. Based on the inequality 14 + t ≤ 25, where t represents the cost of the T-shirt, what is the most Trevor can spend on the T-shirt?
50. The length of the rectangle is greater than its width. Which inequality represents the possible values of x? X x® V £Ó V
A $9 B $10.99 C $11
F x ≤ 17
H x ≥ 17
G x < 17
J x > 17
D $11.50
ALGEBRA For the given value, state whether each inequality is true or false. (Lesson 8-3) 51. x - 5 > 4, x = 9
52. 9 + a ≤ 3, a = -7
d 53. _ ≥ 8, d = 4 2
54. GEOMETRY The perimeter of a rectangle is 24 centimeters. Find the dimensions if the length is 3 more than twice the width. (Lesson 8-2) ALGEBRA Use the Distributive Property to rewrite each expression as an equivalent algebraic expression. (Lesson 3-1) 55. 4(2 + 8) 56. -2(n + 6) 57. 5(x - 3.5) 58. (9 - d)(-3c)
PREREQUISITE SKILL Solve each equation. (Lesson 3-4) 59. -7x = 14
60. -3y = -27
d 61. _ = -6 -3
c 62. _ = 12 -4
Lesson 8-4 Solving Inequalities by Adding or Subtracting
439
CH
APTER
8
Mid-Chapter Quiz Lessons 8-1 through 8-4
Solve each equation. Check your solution.
Graph each inequality on a number line.
(Lesson 8-1)
(Lesson 8-3)
1. 6y + 42 = 4y
12. x < -3
13. y ≥ 5
2. 12x - 19 = 3x + 8
4 14. _ >d 5
15. f < 11.2
3. 7m - 12 = 2.5m + 2 Define a variable and write an equation to find each number. Then solve. (Lesson 8-1) 4. Twice a number is 150 less than 5 times the number. What is the number? 1 5. One fourth of a number plus 3 is _ that 2 number minus 1. What is the number? 6. TESTS Bobby’s score is 5 less than twice Allan’s score. It is also 45 points greater than Allan’s score. What score did the two boys receive? (Lesson 8-1) 7. MULTIPLE CHOICE An online computer game community has two membership plans. The first plan gives you unlimited play time for $40 a month. The second plan charges a monthly access fee of $4.25 plus $2.75 for each hour you play. After how many hours do the two plans cost the same amount? (Lesson 8-1) A 6.6
B 9.0
C 11.2
D 13.0
Solve each equation. Check your solution. (Lesson 8-2)
8. 8(p - 4) = 2(2p + 1) 9. 0.2x - 1.4 = 15.82 - 0.5x 10. b + 2(b + 5) = 3(b - 1) + 13 11. MULTIPLE CHOICE Which of the following 17 ≤ y? graphs represents the inequality -_ (Lesson 8-3)
5
F x {Î Ó £ ä £ Ó Î { x
G £ä ££ £Ó £Î £{ £x £È £Ç £n £ Óä
18. FITNESS The table shows a gym class’s average results for boys and girls participating in the long jump. Gender
Male Female
x {Î Ó £ ä £ Ó Î { x x {Î Ó £ ä £ Ó Î { x
440 Chapter 8 Equations and Inequalities
Distance
17 feet 5 inches 14 feet 3 inches
Cheyenne could jump no farther than 12 inches more than the average distance for males. Write an inequality that gives the possible distances that Cheyenne could jump. (Lesson 8-3) Solve each inequality. Check your solution. (Lesson 8-4) 19. d + 10 ≥ 12 20. c - (-5) < 24 21. 5 < g - 21 22. -32 ≤ 17 + j 23. k - 3 > 7 24. 7 ≤ m + 1 25. MULTIPLE CHOICE Shanté has $50 to spend on a back-to-school outfit. The blouse she wants is $17. Based on the inequality 17 + s ≤ 50 where s is the cost of a skirt, what is the most that Shanté can spend on a skirt? (Lesson 8-4)
A $17 B $33
H J
Write an inequality for each sentence. (Lesson 8-3) 16. More than 35,000 people attended a concert in Toronto. 17. Toby wants to spend no more than 3 hours working on his model car.
C $43 D $50
8-5
Solving Inequalities by Multiplying or Dividing
Main Ideas • Solve inequalities by multiplying or dividing by a positive number. • Solve inequalities by multiplying or dividing by a negative number.
An astronaut in a space suit weighs about 300 pounds on Earth, but only 50 pounds on the Moon because of weaker gravity. weight on Moon
weight on Earth
300
50
>
Weight of Astronaut (lb)
Location Earth
300
Moon
50
Pluto
67
Mars
113
Neptune
407
Jupiter
796
If the astronaut and space suit each weighed half as much, would the inequality still be true? That is, would the astronaut’s weight still be greater on Earth? a. Divide each side of the inequality 300 > 50 by 2. Is the inequality still true? Explain by using an inequality. b. Would the weight of 5 astronauts be greater on Pluto or on Earth? Explain by using an inequality.
Multiply or Divide by a Positive Number The application above demonstrates how you can solve inequalities by using the Multiplication and Division Properties of Inequalities.
Multiplication and Division Properties
Positive Number
Words
When you multiply or divide each side of an inequality by the same positive number, the inequality remains true.
Symbols
For all numbers a, b, and c, where c > 0,
_ 1. if a > b, then ac > bc and _ c > c. a
The inequality c > 0 means that c is a positive number.
b
_ 2. if a < b, then ac < bc and _ c < c. a
Examples
2 -9
4(2) < 4(6)
3 -9 _ >_
8 < 24
1 > -3
3
b
3
These properties are also true for a ≥ b and a ≤ b. Lesson 8-5 Solving Inequalities by Multiplying or Dividing
441
EXAMPLE
Multiply or Divide by a Positive Number
Solve each inequality. Check your solution. a. 8x ≤ 40 8x ≤ 40
Write the inequality.
8x _ _ ≤ 40
Divide each side by 8.
8
8
x≤5
Simplify.
The solution is x ≤ 5. You can check this solution by substituting 5 or a number less than 5 into the inequality. d b. _ >7 2
_d > 7
2 d 2_ > 2(7) 2
d > 14
Write the inequality. Multiply each side by 2. Simplify.
The solution is d > 14. You can check this solution by substituting a number greater than 14 into the inequality. f 1B. _ < -5
1A. 3x > -15
4
Ling earns $8 per hour. Which inequality can be used to find how many hours he must work in a week to earn at least $120? A 8x < 120
B 8x ≤ 120
C 8x > 120
D 8x ≥ 120
Read the Test Item Key Words Before taking a standardized test, review the meanings of phrases like at least and at most.
You are to write an inequality to represent a real-world problem. Solve the Test Item Words Variable
Amount earned per hour
times number of hours is at least the amount earned each week.
Let x represent the number of hours worked.
Inequality 8
·
x
120
≥
The answer is D.
3 2. It takes Alfonzo _ hour to mow a lawn. Which inequality can be used to 4 find the number of lawns he can mow if he works 15 hours per week? 3 3 3 3 F _ x ≤ 15 G _ x ≥ 15 H _ x > 15 J _ x < 45 4
4
Personal Tutor at pre-alg.com
442 Chapter 8 Equations and Inequalities
4
4
Multiply or Divide by a Negative Number What happens when each side of an inequality is multiplied or divided by a negative number? Multiply each side by -1.
Graph 3 and 4 on a number line.
x {Î Ó £ ä £ Ó Î { x
x {Î Ó £ ä £ Ó Î { x
Since 3 is to the left of 4, 3 < 4.
Since -3 is to the right of -4, -3 > -4.
Notice that the numbers being compared switched positions as a result of being multiplied by a negative number. In other words, their order reversed. These and other examples suggest the following properties.
Multiplication and Division Properties Negative Number The inequality c < 0 means that c is a negative number.
Words
When you multiply or divide each side of an inequality by the same negative number, the inequality symbol must be reversed for the inequality to remain true.
Symbols
For all numbers a, b, and c, where c < 0,
_ 1. if a > b, then ac < bc and _ c < c. a
b
_ 2. if a < b, then ac > bc and _ c > c. a
7>1
Examples
b
-4 < 16
16 -4 -2(7) < -2(1) Reverse the symbols. _ > _ -4
-4
1 > -4
-14 < -2
These properties are also true for a ≥ b and a ≤ b.
EXAMPLE
Multiply or Divide by a Negative Number
Solve each inequality and check your solution. Then graph the solution on a number line. x a. _ ≤4
b. -7x > -56
-3
x _ ≤4
-3 x -3 _ ≥ -3(4) -3
x ≥ -12
Write the inequality. Multiply each side by -3 and reverse the symbol. Check this result.
⫺14
y 4
3A. - _ < 3
Extra Examples at pre-alg.com
⫺12
⫺10
⫺8
-56 -7x _ 63 2
5 3 5. _ ≤_ y 7 4
3 1 6. _y ≤ _
(p. 443)
B 13
C 14
HELP
For See Exercises Examples 11–18 1 19, 20, 2 42, 44 21–28 3
D 20
Solve each inequality. Check your solution. Then graph the solution on a number line. 8. -4t > -20
HOMEWORK
4
7. MULTIPLE CHOICE Koto delivers pizzas on weekends. Her average tip is $1.50 for each pizza that she delivers. How many pizzas must she deliver to earn at least $20 in tips? A 10
Example 3
24
9. -8z ≤ -24
2 g 10. 18 > -_ 3
Solve each inequality. Check your solution. 11. 13a ≥ -26
12. -15 ≤ 5b
13. 144 < 12d
14. 15 ≥ 3t
p 15. _ > 5 6
h 16. 7 ≥ _ 14
17. 3m < 33
18. 8z ≤ -24
19. SOCCER Tomás wants to spend less than $100 for a new soccer ball and shoes. The ball costs $24. Write and solve an inequality that gives the amount that Tomás can spend on shoes. 20. ARCADE Montel spends $0.75 every time he plays his favorite video game. Montel has $10. Write and solve an inequality that shows how many times Montel can play the video game. Solve each inequality. Check your solution. Then graph the solution on a number line. 21. -8 ≤ -4w
22. -6a > -78
23. -25t ≤ 400
24. 18 > -2g
y 25. -_ ≥ 2.4 4
n 26. _ ≥ -0.8 -5
x 27. 6 > _ -7
r 28. _ < -2 -2
29. SWIMMING Andrea swims 40 meters per minute, and she wants to swim at least 2000 meters this morning. Write and solve an inequality to find how long she should swim. EXTRA
PRACTIICE
See pages 780, 801. Self-Check Quiz at pre-alg.com
H.O.T. Problems
Solve each inequality. Check your solution. Then graph the solution on a number line. y -0.3
c 30. -5 ≥ -_
31. -19 > _
1 32. -_ x ≥ -9
1 33. -36 < -_ b
y 34. _ < -7 -3
k 35. _ _ -0.4
m 37. _ ≤ 1.2 -7
4.5
3
2
38. OPEN ENDED Write an inequality that can be solved using the Division Property of Inequality, where the inequality symbol is not reversed. 39. CHALLENGE The product of an integer and -7 is less than -84. Find the least integer that meets this condition.
444 Chapter 8 Equation and Inequalities
40. FIND THE ERROR Brittany and Tamika each solved -45 ≥ 9k. Who is correct? Explain your reasoning. Brittany -45 ≥ 9k
Tamika -45 ≥ 9k
-45 9k 9 ≤ 9
-45 ≥ 9k 9 9
-5 ≥ k
-5 ≤ k
41.
Writing in Math
Use the information on page 441 to explain how inequalities can be used in studying space. Illustrate your answer with inequalities that compare the weight of two astronauts on Mars and on the Moon.
42. The solutions for which inequality are represented by the following graph?
44. Which number is NOT a possible length of the rectangle if the area is less than 36 square inches?
Óä £ £n £Ç £È £x £{
x A _ ≤5
-3 x ≥5 B _ -3
x in.
x C _ -5 3
4 in.
43. GRIDDABLE Isabel is putting water into a 20-gallon fish tank using a 2-quart pitcher. How many pitchers of water will she need to fill the tank?
F 6
H 8
G 7
J
9
ALGEBRA Solve each inequality. Check your solution. (Lessons 8-4) 45. -4 + x > 23
46. c + 18 ≤ -2
47. 6 > n - 10
48. CRAFTS It takes Carolyn two hours to complete a cross-stitch pattern. Carolyn can spend no more than fourteen hours cross-stitching. Write an inequality that represents this situation and use it to determine whether Carolyn can complete 8 cross-stitch patterns. (Lesson 8-3) Find each product. Write in simplest form. (Lesson 5-3) 1 3 49. _ · _ 8
4
7
ab 4 52. _ · _
5 1 51. 2_ · -_
3 _ 50. -_ ·5 9
2
2
6
bc
PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 53. 2x + 3 = 9
54. 5a - 6 = 14
55. 3n - 8 = -26
56. _t + 5 = 2 3
57. _c - 1 = 4
d 58. _ + 3 = 19
4
2
Lesson 8-5 Solving Inequalities by Multiplying or Dividing
445
8-6
Solving Multi-Step Inequalities BrainPOP
Main Idea • Solve inequalities that involve more than one operation.
pre-alg.com
More than 10 million Americans are “frequent runners.“ A rule of thumb for training is that you will generally have enough endurance to finish a race that is up to 3 times your average daily distance. a. Write an inequality that represents the relationship between daily average distance and possible race lengths. b. Your average daily run is 2 kilometers. Write and solve an inequality that represents the amount that you need to increase your daily run by to have enough endurance for a 12-kilometer race.
Inequalities with More than One Operation An inequality may involve more than one operation. To solve the inequality, work backward to undo the operations, just as you did in solving multi-step equations.
EXAMPLE
Solve a Two-Step Inequality
Solve 6x + 15 > 9 and check your solution. Graph the solution on a number line.
Common Misconception Do not reverse the inequality sign just because there is a negative sign in the inequality. Only reverse the sign when you multiply or divide by a negative number.
6x + 15 > 9 6x + 15 - 15 > 9 - 15 6x > -6 x > -1 CHECK
Write the inequality. Subtract 15 from each side. Simplify. Mentally divide each side by 6.
6x + 15 > 9 Write the inequality. 6(0) + 15 > 9 Replace x with a number greater than -1. Try 0. 0 + 15 > 9 Simplify. 15 > 9 The solution checks.
Graph the solution, x > -1. 5
4
3
2
1
1
2
3
4
5
1. Solve 8y -2 ≤ 14 and check your solution. Graph the solution on a number line. 446 Chapter 8 Equations and Inequalities Bob Thomas/Getty Images
EXAMPLE
Reverse the Inequality Symbol
Solve 10 - 3a ≤ 25 + 2a and check your solution. Graph the solution on a number line. 10 - 3a ≤ 25 + 2a
Write the inequality.
10 - 3a - 2a ≤ 25 + 2a - 2a Subtract 2a from each side. 10 - 5a ≤ 25
Simplify.
10 - 10 - 5a ≤ 25 - 10 Inequalities Remember that you must reverse the inequality symbol if you multiply or divide each side of an inequality by a negative number.
Subtract 10 from each side.
-5a ≤ 15
Simplify.
15 -5a _ ≥_
Divide each side by -5 and change ≤ to ≥.
-5
-5
a ≥ -3
Simplify.
Check your solution by substituting a number greater than -3. Graph the solution, a ≥ -3. 5 4 3 2 1 0
1
2
3
4 5
2. Solve 5b + 8 ≥ 7b + 2 and check your solution. Graph the solution on a number line.
When inequalities contain grouping symbols, you can use the Distributive Property to begin simplifying the inequality.
RUNNING Refer to the application at the beginning of the lesson. Tammy wants to be able to run at least the standard marathon distance of 26.2 miles. If the length of her current daily runs is about 4 miles, how many miles should she increase her daily run by to meet her goal? Words Variable Inequality
Real-World Link One of the most popular races in America is the Chicago marathon. In 2004, about 33,000 people completed the 26.2-mile race.
3
times
4 miles
plus
amount of increase
is greater than or equal to
desired distance.
Let d = the amount of increase. 3
·
(4
d)
+
3(4 + d) ≥ 26.2
Write the inequality.
12 + 3d ≥ 26.2
Multiply.
≥
3d ≥ 14.2
Subtract 12 from each side.
14.2 3d ≥ _ 3 3
Divide each side by 3.
26.2
d ≥ 4.7 3
Source: chicagomarathon.com
In order to have enough endurance to run a marathon, Tammy should increase the distance of her average daily run by at least 4.73 miles. Extra Examples at pre-alg.com Matthew Stockman/Getty Images
Lesson 8-6 Solving Multi-Step Inequalities
447
3. BUSINESS Banks estimate the value of a business to determine loans and insurance. The formula for the value of a coffee shop is 40% of its annual sales plus the value of its inventory. The value of Holmes Coffee is at least $150,000. Write and solve an inequality to find the annual sales at Holmes Coffee if its inventory is $26,000. Personal Tutor at pre-alg.com
Solve each inequality and check your solution. Then graph the solution on a number line. Example 1 (p. 446)
Example 2 (p. 447)
Example 3 (p. 447)
HOMEWORK
HELP
For See Exercises Examples 10–15 1 16–19 2 20–23 3
1. 3x + 4 ≤ 31
2. 12a - 4 > 20
3. 2n + 5 > 11 – n
4. y + 1 ≥ 4y + 4
5. 16 - 2c < 14
6. 18 ≤ 12 - 2n
7. -3(b - 1) > 18
8. -2(k + 1) ≥ 16
9. MONEY A company pays Dante’s Web site for advertising on the site. The Web site earns $10 per month plus $0.05 each time a visitor to the site clicks on the advertisement. What is the least number of clicks he needs to make $45 per month or more from this advertiser?
Solve each inequality and check your solution. Then graph the solution on a number line. 10. 2x + 8 > 24
11. 6q + 4 ≤ 28
12. 3y - 1 ≤ 5
13. 9t - 5 ≤ -14
14. 3 + 4c > -13
15. 9 + 2p ≤ 15
16. 4 - 3k ≤ 19
17. 16 - 4n > 20
18. -3b + 4 < -2
19. -5a - 8 > 12
20. 2(n + 3) < -4
21. 2(d + 1) > 16
For Exercises 22 and 23, write and solve an inequality. 22. SALES You earn $2 for every magazine subscription you sell plus a salary of $10 each week. How many subscriptions do you need to sell each week to earn at least $40 each week? 23. HIKING You hike along the Appalachian Trail at 3 miles per hour. You stop for one hour for lunch. You want to walk at least 18 miles. How many hours should you expect to spend on the trail? Solve each inequality and check your solution. Then graph the solution on a number line. 24. 3x - 2 > 10 - x
25. c - 1 < 3c + 5
26. 2 + 0.3y ≥ 11
27. 0.5a - 1.4 ≤ 2.1
1 (6 - c) > 5 28. _ 2
m 29. _ +9≥5 2
30. Four times a number less 6 is greater than two times the same number plus 8. For what number or numbers is this true? 448 Chapter 8 Equations and Inequalities
31. One half of the sum of a number and 6 is less than 25. For what numbers is this true? 32. REAL ESTATE A new real estate agent receives a monthly salary of $1500 plus a 3.5% commission on every home sold. For what amount of monthly sales will the agent earn at least $5000? 33. REPAIRS Carl is having a mechanic fix his car. The mechanic said that the job was going to cost at least $375 for parts and labor. If the cost of the parts was $150, and the mechanic charges $60 an hour, how many hours is the mechanic planning on working on the car?
Real-World Career Real Estate Agent Real estate agents help people buying and selling a home. All states require prospective agents to pass a written test, which usually contains a section on mathematics.
For more information, go to pre-alg.com.
EXTRA
PRACTICE
See pages 781, 801. Self-Check Quiz at pre-alg.com
H.O.T. Problems
34. SCHOOL Nate has scores of 85, 91, 89, and 93 on four tests. What is the least number of points he can get on the fifth test to have an average of at least 90? 35. FUND-RAISERS The booster club at Jefferson High School sells football programs for $1 each. The costs to make the programs are $60 for page layout plus $0.20 for printing each program. If they print 400 programs, how many programs must the Club sell to make at least $200 profit? 36. CAR RENTAL The costs for renting a car from Able Car Rental and from Baker Car Rental are shown in the table. For what mileage does Baker have the better deal? Use the inequality 30 + 0.05x > 20 + 0.10x. Explain why this inequality works.
Rental Car Costs Cost per Day
Cost per Mile
Able
$30
$0.05
Baker
$20
$0.10
37. CELL PHONE SERVICES While reviewing prepay phone plans, Miko found that FoneCom charges a $5.35 monthly fee plus $0.10 per minute. Miko currently has BestPhone service at $10 per month plus $0.05 per minute. Miko figures that her monthly bill would be more with FoneCom. For how many minutes per month does she use the phone? 38. TRAVEL Tim is taking the train to Seattle to visit his grandparents. He was given $5.00 to spend on snacks and reading material. Granola bars cost $0.75 each and a newspaper is $1.25. If Tim buys a newspaper, how many granola bars can he get? 39. OPEN ENDED Write a multi-step inequality that can be solved by first adding 3 to each side. 40. CHALLENGE Assume that k is an integer. Solve the inequality 10 - 2|k| > 4. 41. FIND THE ERROR Jerome and Ryan are solving 2(2y + 3) > y + 1. Who is correct? Explain your reasoning. Jerome 2(2y + 3) > y + 1 4y + 6 > y + 1
42.
Ryan 2(2y + 3) > y + 1 4y + 3 > y + 1
Writing in Math Use the information about running found on page 446 to explain how multi-step inequalities are used in running. Lesson 8-6 Solving Multi-Step Inequalities
Doug Martin
449
43. Sandra’s scores on the first five science tests are shown in the table. Which inequality represents the score she must receive on the sixth test to have an average score of more than 88? A s ≥ 86
44. An art teacher wants to buy at least 2 canvases for each student in her painting class. If there are 30 students in the class and if canvases cost $16 per package, what other information is needed to find the amount the teacher should budget for canvases?
Test
Score
B s ≤ 88
1
85
C s < 88
2
84
F The number of art projects planned for the course
D s > 86
3
90
G The cost of paints and brushes
4
95
5
88
H The number of canvases in each package J The total budget available for art supplies
ALGEBRA Solve each inequality. Check your solution. (Lessons 8-4 and 8-5) q 3
45. 6x < -27
46. -5m ≥ -15
47. 8 > _
n 48. _ ≤ -11
49. -9 + k > 20
50. 22 ≤ -15 + y
51. 12 + z ≤ 8
52. 14 ≥ 7 + a
-4
53. SCHOOL If 12 of the 20 students in a class are boys, what percent are boys? (Lesson 6-5) 1 54. Write _ as a percent. (Lesson 6-5) 200
Express each ratio as a unit rate. (Lesson 6-1) 55. $5 for 2 loaves of bread 57. 24 meters in 4 seconds
56. 200 miles on 12 gallons 58. 9 monthly issues for $11.25
GEOMETRY Find the missing dimension in each rectangle. (Lesson 3-8) 59.
18.4 ft
ᐉ
60.
w 5.1 m
Area = 30.6 m2
Perimeter = 49.6 ft
Math and Recreation Just for Fun It is time to complete your project. Use the information and data you have gathered about
recreational activities to prepare a Web page or poster. Be sure to include a scatter plot and a prediction for each activity. Cross-Curricular Project at pre-alg.com
450 Chapter 8 Equations and Inequalities
CH
APTER
8
Study Guide and Review
wnload Vocabulary view from pre-alg.com
Key Vocabulary
Be sure the following Key Concepts are noted in your Foldable.
Key Concepts Solving Equations
(Lessons 8-1 and 8-2)
• Use the Addition or Subtraction Property of Equality to isolate the variables on one side of an equation.
identity (p. 426) inequality (p. 430) null or empty set (p. 426)
Vocabulary Check Determine whether each statement is true or false. If false, replace the underlined word or phrase to make a true statement. 1. When an equation has no solution, the solution set is the null set.
• Use the Distributive Property to remove the grouping symbols.
2. The inequality n + 8 - 8 ≥ 14 - 8 demonstrates the Subtraction Property of Inequality.
Solving Inequalities
3. An equation that is true for every value of the variable is called an inequality.
(Lessons 8-3 to 8-6)
• An inequality is a mathematical sentence that contains , ≤, or ≥.
4. The inequality x (4) < 7(4) demonstrates 4 the Division Property of Inequality.
• Solving an inequality means finding values for the variable that make the inequality true.
5. A mathematical sentence that contains , ≤, or ≥ is called an empty set.
• When you multiply or divide each side of an inequality by a positive number, the inequality symbol remains the same.
6. When the final result in solving an equation is 5 = -8, the solution set is the null set.
• When you multiply or divide each side of an inequality by a negative number, the inequality symbol must be reversed.
7. The symbol ≥ means is less than or equal to.
• To solve an inequality that involves more than one operation, work backward to undo the operations.
Vocabulary Review at pre.alg.com
Chapter 8 Study Guide and Review
451
CH
A PT ER
8
Study Guide and Review
Lesson-by-Lesson Review 8–1
Solving Equations with Variables on Each Side
(pp. 420–423)
Solve each equation. Check your solution.
Example 1 Solve 7x = 3x - 12.
8. 2a + 9 = 5a
7x = 3x - 12 Write the equation. 7x - 3x = 3x - 3x - 12 Subtract 3x from each
9. x - 4 = 3x
10. 3y - 8 = y
11. 19t = 26 + 6t
12. 12 + 1.5x = 9x
13. 5b - 1 = 2.5b - 4
side.
4x = -12
Simplify.
4x -12 _ =_
Divide each side by 4.
4
14. CONCERTS An outdoor concert venue is planning on increasing the number of 1 for next year. This will concerts by 14 increase their number of concerts by 3. How many concerts will they host this year?
4
x = -3
Simplify.
15. An online DVD rental club has two membership plans as shown. In how many months would the total cost of the two plans be the same?
A
Membership Fee $20
Cost Per Month $5
B
$30
$3
Plan
8–2
Solving Equations with Grouping Symbols
(pp. 424–428)
Solve each equation. Check your solution. 16. 4(k + 1) = 16 17. 2(n - 5) = 8 18. 11 + 2q = 2(q + 4) 3 1 19. _ (t + 8) = _ t 2
4
20. 4(x + 2.5) = 3(7 + x) 21. 3(x + 1) - 5 = 3x - 2 22. GEOMETRY The perimeter of a rectangle is 84 meters. Find the dimensions of the rectangle if the length is 3 meters less than twice the width.
452 Chapter 8 Equations and Inequalities
Example 2 Solve 2(x + 3) = 15. 2(x + 3) = 15 2x + 6 = 15 2x + 6 - 6 = 15 - 6 2x = 9 9 2x _ =_ 2 2
x = 4.5
Write the equation. Use the Distributive Property. Subtract 6 from each side. Simplify. Divide each side by 2. Simplify.
Mixed Problem Solving
For mixed problem-solving practice, see page 801.
8–3
Inequalities
(pp. 430–434)
For the given value, state whether each inequality is true or false.
Example 3 State whether n + 11 < 14 is true or false for n = 5.
23. x + 4 > 9, x = 12 25. 6r > 30, r = 5
n + 11 < 14 Write the inequality. 5 + 11 < 14 Replace n with 5. 16 ≮ 14 Simplify.
26. 15 ≤ 5n, n = 8
The sentence is false.
24. 12 - t < 5, t = 3
27. 3n + 1 ≥ 14, n = 7 28. 23 ≤ _c + 2, c = 10 4
29. CAMPING When camping, Stephán and his friends usually use at least 3 logs for fire each night. Write an inequality that represents this situation. 30. DIVING In a diving competition, the diver in first place has a total score of 345.4. Ming has scored 68.2, 68.9, 67.5, and 71.7 for her first four dives and has one more dive remaining. Write an inequality to show the score x that Ming must receive on her fifth dive in order to overtake the diver in first place.
8–4
Solving Inequalities by Adding or Subtracting
(pp. 435–439)
Solve each inequality. Then graph the solution on a number line.
Example 4 Solve x - 7 ≤ 3. Then graph the solution on a number line.
31. b - 9 ≥ 8
32. 15 > 3 + n
33. x + 4.8 ≤ 2
34. r + 5.7 ≤ 6.1
1 35. t + _ 19 47. 5n + 4 ≤ 24 48. 6 ≥ _r + 1 7
t 49. _ + 15 < 21 -2
50. 3(a + 8.4) > 30 1 51. _ + 2b < 13 + 5b 4
52. SALES A car sales associate receives a monthly salary of $1700 a month plus 8% commission on every car sold. For what amount of monthly sales will the sales associate earn at least $4200?
454 Chapter 8 Equations and Inequalities
Example 6 Solve 4t + 7 < -5. Write the inequality. 4t + 7 < -5 4t + 7 - 7 < -5 - 7 Subtract 7 from each side. 4t < -12 Simplify. t < -3 Mentally divide each side by 4.
CH
A PT ER
8
Practice Test
Solve each equation. Check your solution.
Write an inequality for each graph.
1. 7x - 3 = 10x 2. p - 9 = 4p
14.
3. 2(6 - 5d) = -8
15.
4. 4(a + 3) = 20
16.
⫺2 ⫺6
5. 2.3n - 8 = 1.2n + 3
0 ⫺4
2 ⫺2
4
6
2
4
£ ä £ Ó Î { x È Ç n
3 5 6. _ y-5=_ y-3 8 8
7. 6 + 2(x - 4) = 2(x - 1) 1 8. _ (9b + 1) = b - 1 3
9. MULTIPLE CHOICE For a project, a class is divided into two groups and each group has to make a video. Group A’s video is 20 seconds less than twice the length of Group B’s video. Group A’s video is also 255 seconds longer than Group B’s video. Which equation represents this information? A 2a + 20 = b + 255
17. SALES The Cookie Factory has a fixed cost of $300 per month plus $0.45 for each cookie sold. Each cookie sells for $0.95. How many cookies must be sold during one month for the profit to be at least $100? 18. MULTIPLE CHOICE Danny earns $8.50 per hour working at a movie theater. Which inequality can be used to find how many hours he must work each week to earn at least $100 a week? F
B 2b - 20 = b + 255
8.50h < 100
H 8.50h ≤ 100
G 8.50h > 100
C 20b - 255 = b + 20
J
8.50h ≥ 100
D a + 255 = b - 20 For Exercises 10–12, define a variable and write an equation to find each number. Then solve. 10. Eight more than three times a number equals four less than the number. 11. The product of a number and five is twelve more than the number. 12. GEOMETRY The perimeter of the rectangle is 22 feet. Find the dimensions of the rectangle. w 2w + 3.5
13. SHOPPING The cost of purchasing four shirts is at least $120. Write an inequality to describe this situation.
Chapter Test at pre-alg.com
Solve each inequality and check your solution. Then graph the solution on a number line. 19. -4 ≥ p - 2
20. 3x ≥ 15
21. -42 < -0.6x
22. c - 3 ≤ 4c + 9
23. 7(3 - 2b) ≥ 5b + 2
1 1 24. _ (a + 4) > _ (a - 8) 2
4
25. MULTIPLE CHOICE The Lapeer Nature Club wants to raise at least $4000 for conservation. They have been given a $150 dollar donation and are selling canvas bags for $55 each to raise the rest of the money. Which inequality describes how many bags they need to sell in order to reach this goal? A x ≥ 35 B x ≤ 35 C x ≤ 70 D x ≥ 70
Chapter 8 Practice Test
455
CH
A PT ER
8
Standardized Test Practice Cumulative, Chapters 1–8
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. The cost, c, of renting a moving truck can be found using the equation c = 20.75 + 31.50d, where d is the number of days you rent the truck. What would be the total cost of renting a truck for 4 days? C $158.50 A $126.00 B $146.75 D $172.25
4. A sequence of numbers was generated using the rule 3n - 1, where n represents a number’s position in the sequence. Which sequence fits this rule? A 1, 3, 5, 7, 9, … B 1, 4, 7, 10, 13, … C 2, 4, 6, 8, 10, … D 2, 5, 8, 11, 14, … 3 4 5. Which fraction is between _ and _ ? 2 F _
4
5
3
2. A music store surveyed 100 of its customers about their preferred styles of music. The results of the survey are shown in the survey.
5 G _ 7 19 H _ 25
7 J _ 8
Favorite Style of Music Style Country
Frequency 25
Rock
38
Jazz
18
Classical
12
Other
7
If the store only uses these data to order new CDs, what conclusion can be drawn from the data? F More than half of each order should be country and rock CDs. G More than half of each order should be rock CDs. H Only country, rock, and jazz CDs should be ordered. J About a fourth of each order should be classical music CDs.
3. GRIDDABLE Find the next term in the pattern below. 1, 3, 7, 13, 21, 31, … 456 Chapter 8 Equations and Inequalities
6. GRIDDABLE Colleen is using a punch recipe that calls for 12 ounces of fruit juice for every 40 ounces of lemon-lime soda. If she uses 60 ounces of lemon-lime soda, how many ounces of fruit juice will she need? 7. Robert, Isabelle, Michael, and Katrina are going to a football game. The total cost of the tickets is $231.75. Robert paid $60, Isabelle 1 of paid 20% of the total cost, Michael paid _ 4 the total cost, and Katrina paid the rest. Who paid the greatest amount? A Robert B Isabelle C Michael D Katrina 8. A couch is on sale for 20% off the regular price of $480. How much money is discounted off the regular price? F $384 G $362 H $96 J $84 Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
9. Kelly’s deck has an area of 660 square feet.
12. The cost, c, of renting a car can be found using the equation c = 50 + 0.10m, where m is the number of miles you drive the car. What would be the total cost of renting a car and driving it 200 miles? H $50 F $20 G $0.10 J $70
FT
What is the length of the deck if the width is 11 feet? A 66 ft B 60 ft C 50 ft D 45 ft
13. The cost, c, of renting a tent site can be found using the equation c = 12.50 + 2.50p, where p is the number of people you will have on the site. What would be the total cost of renting a tent site for 5 people? C $25.00 A $12.50 B $15.00 D $65.00
10. Mary Ann saved $56 when she purchased a television on clearance at an electronics store. If the sale price was 20% off the regular price, what was the regular price? F $250 G $260 H $275 J $280
If you get finished with the test before the end of the time allowed, go back and check your work.
Pre-AP Record your answers on a sheet of paper. Show your work. 14. Kevin earns a monthly salary of $1750. In addition to his salary, he receives a $250 bonus for every car that he sells. He wants to earn at least $3000 per month. a. Write an inequality to represent this situation. b. Solve the inequality that you found in part a. c. What is the minimum number of cars he must sell? d. How many cars will he have to sell if he wants to earn at least $4000 per month?
11. The cost, c, of hiring a plumber can be found using the equation c = 75 + 40h, where h is the number of hours the plumber worked. What would be the total cost of hiring a plumber for 3 hours? A $40 B $75 C $195 D $120
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Go to Lesson...
8-1
6-10
1-1
3-7
5-2
6-2
5-2
6-6
3-8
6-8
8-1
8-1
1-5
8-5
Chapter 8 Standardized Test Practice
457
Applying Algebra to Geometry Focus Analyze figures in two- and three-dimensional space. Find areas of two-dimensional figures and volumes and surface areas of three-dimensional figures.
CHAPTER 9 Real Numbers and Right Triangles Use different forms of numbers appropriate for different situations. Use indirect measurement to solve problems.
CHAPTER 10 Two-Dimensional Figures Use transformational geometry to develop spatial sense. Use geometry to model and describe the physical world.
CHAPTER 11 Three-Dimensional Figures Determine measures of three-dimensional figures. Describe how changes in dimensions affect area and volume measures. 458 Unit 4 Applying Algebra to Geometry Bill Ross/CORBIS
Algebra and Architecture Able to Leap Tall Buildings The tallest building in the United States is the Sears Tower in Chicago. It has a height of 1450 feet. Did you know that at 790 feet tall, the John Hancock Tower in Boston is the tallest building in Massachusetts? In this project, you will be exploring how geometry and algebra can help you describe unusual or large structures in the United States and from around the world. Log on to pre-alg.com to begin.
Unit 4 Applying Algebra to Geometry
459
Real Numbers and Right Triangles
9 •
Identify numbers in the real number system.
•
Use the Pythagorean Theorem, the Distance Formula, and the Midpoint Formula.
•
Classify angles and triangles and identify and use properties of similar figures.
•
Use trigonometric ratios to solve problems.
Key Vocabulary hypotenuse (p. 485) irrational numbers (p. 469) real numbers (p. 469) similar figures (p. 497)
Real-World Link Roller Coasters A rider at the top of the Titan roller coaster, 255 feet above the ground, can see for approximately 19.5 miles on a clear day.
Real Numbers and Right Triangles Make this Foldable to help you organize information about real numbers and right triangles. Begin with three plain sheets of 812“ × 11“ paper.
1 Fold to make a triangle. Cut off the extra paper.
3 Stack the three squares and staple along the fold.
460 Chapter 9 Real Numbers and Right Triangles Courtesy of Six Flags Over Texas
2 Repeat Step 1 twice. You now have three squares.
4 Label each section with a topic.
3IGHT S 5RIANGLE
GET READY for Chapter 9 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Replace each with , or = to make a true statement. (Prerequisite Skills, p. 742) 1. 3.2
2. 7.8
3.5
3. 5.13
5.16
4. 4.92
5. 2.62
2.6
6. 3.4
7. 0.07
0.7
8. 1.16
7.7 4.89 3.41 1.06
Example 1 Replace with , or = to make 29.19 29.2 a true statement.
29.19
Line up the decimal points.
29.2
The digits in the tenths place are not the same.
1 tenth < 2 tenths, so 29.19 < 29.2.
9. FIELD HOCKEY The following shows the winning percents of five field hockey teams. Order them from greatest to least. 0.523, 0.546, 0.601, 0.594, 0.509 (Prerequisite Skills, p. 742)
Solve each equation. (Lesson 3-4) 10. 3x = 24
11. 7y = 49
12. 120 = 2n
13. 54 = 6a
14. 90 = 10m 16. 15d = 165
Example 2 Solve 14w = 56.
14w = 56
Write the equation.
15. 144 = 12m
14w 56 _ =_
Divide each side by 14.
17. 182 = 14w
w=4
14
14
Simplify.
18. COOKIES A batch of cookies contains 8 tablespoons of sugar. How many batches of cookies contain 64 tablespoons of sugar?
19. (3 - 1)2 + (4 - 2)2
Example 3 Evaluate (6 - 2)2 + (9 - 7)2.
20. (5 - 2)2 + (6 - 3)2
(6 - 2)2 + (9 - 7)2 = 42 + 22
Evaluate each expression. (Lesson 4-1)
21. (4 - 7)2 + (3 - 8)2 22. (8 - 2)2 + (3 - 9)2
Simplify the expressions inside parentheses first.
23. (2 - 6)2 + [(-8) - 1]2
= 16 + 4
Evaluate 42 and 22.
24. (-7 - 2)2 + [3 - (-4)]2
= 20
Simplify.
25. BIOLOGY Suppose a virus splits into two viruses every 45 minutes. How many viruses are there after 5 hours 15 minutes? (Lesson 4-1)
Chapter 9 Get Ready for Chapter 9
461
EXPLORE
9-1
Algebra Lab
Squares and Square Roots Numbers raised to the second power are called squares. You can use a geometric model to discover the reason for the term.
ACTIVITY 1 Use algebra tiles to evaluate 62. • The expression 62 is the product 6 × 6. Products can be represented by squares with one factor as the length and the other as the width. • Arrange tiles in a 6-by-6 square. • Since 6 × 6 = 36, 62 = 36.
ANALYZE THE RESULTS Model each power. Apply what you learned to evaluate it. 1. 32
2. 52
3. 72
4. 82
5. 102
6. 122
7. Explain why n2 is often called n squared. The opposite of squaring a number is finding a square root. To find the square root of a number, find two equal factors whose product is that number. The positive square root of a number is the principal square root. The symbol for the principal square root is √.
ACTIVITY 2 Use algebra tiles to find √ 25 . • You know that a square number can be represented by the area of a square. To find the square root of 25, arrange 25 tiles into a square. • 25 tiles can be arranged in a 5-by-5 square. Therefore, 25 = 5 × 5, or 52. • The principal square root of 25 is 5.
ANALYZE THE RESULTS Model each square root. Apply what you learned to evaluate it. 8. √ 4
9. √ 16
10. √ 81
11. √49
12. √ 100
13. √ 121
14. What part of the model represents the square root of the area of the square? 462 Chapter 9 Real Numbers and Right Triangles
Suppose you try to arrange 50 tiles into a square. You discover that it’s impossible. This suggests that 50 is not a perfect square. You can estimate the square roots of numbers that are not perfect squares.
ACTIVITY 3 Use algebra tiles to estimate the principal square root of 50. • Arrange 50 tiles into the largest square possible. The largest square possible has 49 tiles, with one left over.
Ȗ{ Ç
• Add tiles until you have the next larger square. You need to add 7 tiles on top and 7 tiles on the side, and then the leftover tile from above can be placed in the upper right corner. Therefore, you added 14 new tiles to make a square that has 64 tiles.
ȖÈ{ n
• The square root of 49 is 7 and the square root of 64 is 8. Therefore the square root of 50 is between the whole numbers 7 and 8. Since 50 is closer to 49 than 50 is closer to 7 than 8. 64, you can expect that √ Ȗ{ Ȗxä Ç
ȖÈ{ n
• Verify the estimate with a calculator. 2nd [ √] 50 ENTER 7.071067812
ANALYZE THE RESULTS Model each square root. Apply what you learned to estimate the principal square root. 15. 20
16. 44
17. 58
18. 69
19. 94
20. 111
21. MAKE A CONJECTURE Describe a method that could be used to find the square root of a number by squaring numbers to estimate, rather than by taking square roots to estimate. Explore 9-1 Algebra Lab: Squares and Square Roots
463
9-1
Squares and Square Roots
Main Ideas • Find squares and square roots. • Estimate square roots.
New Vocabulary perfect square square root radical sign
Values of x2 are shown in the second column in the table. Guess and check to find the value of x that corresponds to x2. If you cannot find an exact answer, estimate with decimals to the nearest tenth to find an approximate answer.
x
x2 25 49 169 225
a. Describe the difference between the first four and the last four values of x.
8 12
b. Explain how you found an exact answer for the first four values of x.
65 110
c. How did you find an estimate for the last four values of x?
Squares and Square Roots Numbers like 25, 49, 169, and 225 are perfect squares because they are squares of integers. 5 × 5 or 52
7 × 7 or 72
13 × 13 or 132
15 × 15 or 152
25
49
169
225
A square root of a number is one of two equal factors of the number. Every positive number has a positive square root and a negative square root. A negative number like –9 has no real square root because the square of a number cannot be negative. Square Root Words
A square root of a number is one of its two equal factors.
Symbols
If x2 = y, then x is a square root of y.
Example
Since 5 · 5 or 52 = 25, 5 is a square root of 25. Since (-5) · (-5) or (-5)2 = 25, -5 is a square root of 25.
A radical sign, √, is used to indicate a positive square root. Since every positive number has both a positive and a negative square root, different notations are used to indicate one or both square roots.
EXAMPLE
Find Square Roots
Find each square root.
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
a. √36
√ 36 indicates the positive square root of 36.
Since 62 = 36, √ 36 = 6. b. - √ 81 - √ 81 indicates the negative square root of 81. Since 92 = 81, - √ 81 = -9.
464 Chapter 9 Real Numbers and Right Triangles
Reading Math Plus or Minus Symbol The notation ± √ 9 is read plus or minus the square root of 9.
c. ± √ 9 ± √9 indicates both square roots of 9. Since 32 = 9, √ 9 = 3 and - √9 = -3. d. √ x2 √ x2 indicates the positive square root of x2. x may be negative, but ⎪x⎥ is positive, so √ x2 = ⎪x⎥.
1A. √ 49
Reading Math Approximately Equal to Symbol The symbol ≈ is read is approximately equal to.
1C. ± 兹 144
1B. - √ 100
EXAMPLE
1D.
y2 √
Find Square Roots with a Calculator
Use a calculator to find each square root to the nearest whole number. a. √10 2nd [ √] 10 ENTER 3.16227766
Use a calculator.
10 ≈ 3.2 兹
Round to the nearest tenth.
CHECK Since (3)2 = 9, the
10
answer is reasonable. 1
2
3
4
5
6
7
8
9
10
b. - √ 27 2nd [ √] 27 ENTER 5.19615242
Use a calculator.
√ 27 ≈ -5.2
Round to the nearest tenth.
CHECK Since (-5)2 = 25 , the
⫺27
answer is reasonable. ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
1
2
3
2A. √ 14
2B. - √ 12
Estimate Square Roots You can also estimate square roots mentally by using perfect squares.
EXAMPLE Appropriate Forms of Numbers Express a number as a square root if an exact answer is needed. Express a number as a decimal if an approximation is needed.
Estimate Square Roots
Estimate each square root to the nearest integer. a. √ 38 • The first perfect square less than 38 is 36. • The first perfect square greater than 38 is 49.
√ 36 = 6 √ 49 = 7
• Plot each square root on a number line. ȖÎÈ ȖÎn È
Ȗ{ Ç
The square root of 38 is between the whole numbers 6 and 7. Since 38 38 is closer to 6 than 7. is closer to 36 than 49, you can expect that √ Extra Examples at pre-alg.com
Lesson 9-1 Squares and Square Roots
465
b. - √ 175 • The first perfect square less than 175 is 169. • The first perfect square greater than 175 is 196. • Plot each square root on a number line.
√ 169 = 13 √ 196 = 14
Ȗ£È Ȗ£Çx Ȗ£È £{
£Î
The negative square root of 175 is between the integers -13 and -14. Since 175 is closer to 169 than 196, you can expect that - √ 175 is closer to -13 than -14.
3A. √ 120
3B. - √ 18
When finding square roots in real-world situations, use the positive, or principal, square root when a negative answer does not make sense.
Real-World Link To estimate how far you can see from a point above the horizon, you can use the formula D = 1.22 × 兹 A where D is the distance in miles and A is the altitude, or height, in feet.
SCIENCE Use the information at the left. The light on Cape Hatteras Lighthouse in North Carolina is 208 feet high. On a clear day, from about what distance on the ocean is the light visible? Round to the 196 = 1 ⫻ 14 = 14 nearest tenth. Estimate 1 ⫻ √ D = 1.22 × √ A Write the formula. √ = 1.22 × 208 Replace A with 208. ≈ 17.5951 Evaluate the square root first. Then multiply. On a clear day, the light will be visible from about 17.6 miles.
4. SCIENCE Spring Port Ledge Lighthouse in Maine stands roughly 55 feet high. Estimate and then find about how far a person who is standing on the observation deck can see on a clear day. Round to the nearest tenth. Personal Tutor at pre-alg.com
Example 1 (pp. 464–465)
Example 2 (p. 465)
Example 3
Find each square root. 25 1. √
2. - √ 64
3. ± √ 36
Use a calculator to find each square root to the nearest tenth. 4. √15
5. - √ 32
Estimate each square root to the nearest integer. Do not use a calculator.
(pp. 465–466)
6. √66
Example 4
8. BASEBALL A baseball diamond is actually a square with an area of 8100 square feet. The Cincinnati Reds cover their diamond with a tarp to protect it from the rain. The sides are all the same length. How long is the tarp on each side?
(p. 466)
466 Chapter 9 Real Numbers and Right Triangles Owaki-Kulla/CORBIS
7. - √ 103
HOMEWORK
HELP
For See Exercises Examples 9–14 1 15–22 2 23–28 3 29, 30 4
Find each square root. 9. √16 12. - √ 25
10. √ 49
11. - √1
13. ± √ 100
14. ± √ 196
Use a calculator to find each square root to the nearest tenth. 15. √ 30
16. √ 56
17. - √ 43
18. - √ 86
19. √180
20. √ 250
21. ± √ 0.75
22. ± √ 3.05
Estimate each square root to the nearest integer. Do not use a calculator. 79 23. √
24. √ 95
25. - √ 54
26. - √ 125
27. ± √ 200
28. ± √ 396
ROLLER COASTERS For Exercises 29 and 30, use the table shown and refer to Example 4 on page 466. 29. On a clear day, estimate how far a person can see from the top hill of the Vortex. Then calculate the distance. 30. Estimate how far a person can see on a clear day from the top hill of the Titan. Then calculate the distance.
Coaster
Maximum Height (ft)
Double Loop The Villain Mean Streak Raptor The Beast Vortex Titan Shockwave
95 120 161 137 110 148 255 116
Source: Roller Coaster Database
31. RESEARCH Use the Internet or another source to find the tallest roller coaster in the world. How far would you be able to see from the top of the roller coaster on a clear day? Complete Exercises 32–36 mentally. 54 lies between which two consecutive whole numbers? 32. The number √ 65 or 9? Explain your reasoning. 33. Which is greater, √ 120 ? Explain your reasoning. 34. Which is less, 11 or √ 35. Find a square root that lies between 17 and 18. 77 , −8, − √ 83 , 9, −10, − √ 76 , √ 65 from least to greatest. 36. Order √ GEOMETRY The area of each square is given. Estimate the length of a side of each square to the nearest tenth. Then find its approximate perimeter. 37.
38. 109 in2
39. 203 cm 2
70 m 2
40. Find the negative square root of 1000 to the nearest tenth. EXTRA PRACTICE See pages 781, 802. Self-Check Quiz at pre-alg.com
, what is the value of x to the nearest tenth? 41. If x = √5000 42. CONSTRUCTION City code requires that a reception hall must allow 4 square feet for each person on the dance floor. The reception hall wants to have a dance floor that is a square and that is large enough for 100 people at a time. What is the length of each side of the dance floor? Lesson 9-1 Squares and Square Roots
467
43. GEOMETRY Estimate the perimeter of a square that has an area of 2080 square meters. Then calculate the perimeter. Round to the nearest tenth.
H.O.T. Problems
44. OPEN ENDED Write a number for which the negative square root is not an integer. Then graph the negative square root. 45. NUMBER SENSE What are the possibilities for the ending digit of a number that has a whole number square root? Explain your reasoning. CHALLENGE For Exercises 46–48, use the following information. Squaring a number and finding the square root of a number are inverse operations. That is, one operation undoes the other operation. Use inverse operations to evaluate each expression. 64 )2 46. ( √
47. ( √ 100 )2
48. ( √ 169 )2
49. REASONING Use the pattern from Exercises 46–48 to find ( √a)2 if a ≥ 0. 50.
Writing in Math How are square roots related to factors? Give an example of a number between 100 and 200 whose square root is a whole number and an example of a number between 100 and 200 whose square root is a decimal that does not terminate.
51. Which point on the number line best ? represents √210 !
"
£Î°x £Î°Çx
A A
#
$
£{ £{°Óx £{°x £{°Çx
B B
CC
52. The area of each square is 25 square units. Find the perimeter of the figure.
£x
D D
F 60 units
H 100 units
G 75 units
J 125 units
Solve each inequality. (Lessons 8-5 and 8-6) 53. 4y > 24
a 54. _ < -7
55. 18 ≥ -2k
56. 2x + 5 < 17
57. 2t - 3 ≥ 1.4t + 6
58. 12r - 4 > 7 + 12r
0.3
59. SALES Ice cream sales increase as the temperature outside increases. Describe the slope of a line of fit that represents this situation. (Lesson 7-8) 60. Determine whether the relation (4, -1), (3, 5), (-4, 1), (4, 2) is a function. Explain. (Lesson 7-1) 61. TRAVEL Martin drives 6 hours at an average rate of 65 miles per hour. What is the distance Martin travels? Use d = rt. (Lesson 5-3)
PREREQUISITE SKILL Explain why each number is a rational number. (Lesson 5-2) 10 1 62. _ 63. 1_ 64. 0.75 65. 0.8 66. 6 67. -7 2
2
468 Chapter 9 Real Numbers and Right Triangles
9-2
The Real Number System
Main Ideas • Identify and compare numbers in the real number system.
In this activity, you will find the length of a side of a square that has an area of 2 square units.
• Solve equations by finding square roots.
a. The small square at the right has an area of 1 square unit. Find the area of the shaded triangle.
New Vocabulary
b. Suppose four squares are arranged as shown. What shape is formed by the shaded triangles?
irrational numbers reall numbers b
c. Find the total area of the four shaded triangles. d. What number represents the length of the side of the shaded square?
Identify and Compare Real Numbers Rational numbers can be written as fractions. A few examples of rational numbers are given below. 2 -6 8_ 0.05 -2.6 5. 3 -8.12121212… 5
√ 16
Not all numbers are rational numbers. A few examples of numbers that are not rational are given below. These numbers are not repeating or terminating decimals. They are called irrational numbers. Look Back
= 3.14159…
0.101001000100001…
√2 =1.414213562…
To review rational numbers, see Lesson 5-2.
Irrational Number An irrational number is a number that cannot be expressed as _, where a b and b are integers and b does not equal 0. a
The set of rational numbers and the set of irrational numbers together make up the set of real numbers. The Venn diagram shows the relationship among the real numbers. Real Numbers
1 3
0.2
Rational Numbers 0.6 Whole Numbers
Integers ⫺4
⫺3
0 2 5
0.010010001...
⫺8 2
⫺3
0.25
Irrational Numbers
2
Natural Numbers
Lesson 9-2 The Real Number System
469
EXAMPLE
Classify Real Numbers
Name all of the sets of numbers to which each real number belongs. 3 a. 0.
This repeating decimal is a rational number because it is 1 equivalent to _ . 1 ÷ 3 = 0.33333…
b. √ 67
√ 67 = 8.185352772… It is not the square root of a perfect square so it is irrational.
28 c. -_
28 Since -_ = -7, this number is an integer and a rational number.
10 d. _
10 Since _ = 2.5, this number is a terminating decimal and thus a 4 rational number.
3
4
4
4
1A. 0.7
1B. - √ 121
9 1C. _
1D. 9
5
EXAMPLE
Compare Real Numbers on a Number Line 3 5_ a true statement.
with , or = to make √ 34
a. Replace
8
Express each number as a decimal. Then graph the numbers. 5 3 = 5.375
√ 34 = 5.830951895…
8
3
58 5.0
Look Back To review comparing fractions and decimals, see Lesson 5-1.
5.1
5.2
5.3
5.4
34 5.5
5.6
5.7
5.8
5.9
6.0
3 √ 3 Since √ 34 is to the right of 5 _ , 34 > 5 _ . 8
8
1 √ , 17 , 4.4, and √ 16 from least to greatest. b. Order 4 _ 2
Express each number as a decimal. Then compare the decimals. 1 4_ = 4.5 4. 4 = 4.444444444… 2
√ 17 = 4.123105626…
√ 16 = 4 1
16 17 4.0
4.1
4.4 4.2
4.3
4.4
42 4.5
4.6
4.7
4.8
4.9
5.0
1 , √17 , 4. From least to greatest, the order is √16 4, 4 _ . 2 3 √ 2A. Replace with , or = to make 7 _ 58 a true statement. 5 12 1 _ _ 12 , 3.3, , and 3 from greatest to least. 2B. Order √ 3
3
Solve Equations by Finding Square Roots Some equations have irrational number solutions. Just as you can solve an equation by adding the same number to each side, you can solve certain equations by taking the square root of each side. 470 Chapter 9 Real Numbers and Right Triangles
EXAMPLE
Solve Equations
Solve each equation. Round to the nearest tenth, if necessary. a. x2 = 64 x2 = 64 √x2 = √ 64
Write the equation. Take the square root of each side.
x = √ 64 or x = - √ 64 Find the positive and negative square root. x=8
or x = -8
The solutions are 8 and -8. b. 2n2 = 170 Check Reasonableness Check the results by evaluating 92 and (-9)2.
2n2 = 170 n2
= 85
Divide each side by 2.
√ n2 = √ 85
Take the square root of each side.
n = √ 85 or n = - √ 85 Find the positive and negative square root.
92 = 81 (-9)2 = 81 Since 81 is close to 85, the solutions are reasonable.
Write the equation.
n ≈ 9.2 or n ≈ -9.2
Use a calculator.
The solutions are 9.2 and -9.2.
3A. 363 = 3d2
3B. y2 = 30
Personal Tutor at pre-alg.com
HANG GLIDING The aspect ratio of a hang glider allows it to glide 2 through the air. The formula for the aspect ratio R is R = s, where A s is the wingspan and A is the area of the wing. What is the wingspan of a hang glider if its aspect ratio is 4.5 and the area of the wing is 50 square feet? 2
Real-World Link The record for hang gliding distance belongs to Mike Barber who hang glided 437 miles in Zapata, Texas, in June 2002.
s R=_
A 2 _ 4.5 = s 50
225 = s2 √ 225 =
√ s2
15 = s
Write the formula. Replace R with 4.5 and A with 50. Multiply each side by 50. Take the positive square root of each side. Simplify.
The wingspan of the hang glider is 15 feet.
4. SEISMIC WAVES A tsunami is caused by an earthquake on the ocean floor. s2 = 9.61, where The speed of a tsunami can be measured by the formula _ d s is the speed of the wave in meters per second and d is the depth of the ocean in meters where the earthquake occurs. What is the speed of a tsunami if an earthquake occurs at a depth of 632 meters? Round to the nearest tenth. Extra Examples at pre-alg.com G. Kalt/zefa/CORBIS
Lesson 9-2 The Real Number System
471
Example 1 (p. 470)
Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. 1. 7
Example 2 (p. 470)
3 3. -_
2. 0.5555…
with , or = to make a true statement. 6. - √ 74 -8.4
Replace each 4 5. 6_ 5
4. √ 12
4
√ 48
Order each set of numbers from least to greatest. 10 1 7. 3.7, 3 3, √ 13 , _ 8. √ 110 , 10_ , 10. 5, 10.15 5
Example 3 (p. 471)
Example 4 (p. 471)
HOMEWORK
HELP
For See Exercises Examples 12–23 1 24–33 2 34–41 3 42–43 4
3
5
ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 9. y2 = 25
10. 3m2 = 222
11. LANDSCAPING A sprinkler waters a circular area of the lawn as shown. The formula A = 3.14r2 measures the distance r the sprinkler shoots water within a circular area A. How far is the sprinkler shooting water if it waters an area of 572.6 square feet? Round to the nearest tenth.
R
Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. 1 13. _
2 14. _
15. 4
24 16. -_
17. 7.6
18. - √ 64
19. 0.131313…
20. 2.8
56 21. -_
22. 0. 2
23. - √ 100
12. 8
2
8
8
with , or = to make a true statement.
Replace each 1 24. 5_ 4
5
√ 26
27. - √ 18
25. √ 80
3 -4_
1 28. 1_
8
2
26. -3.3
9.2 √ 2.25
- √ 10
29. - √ 6.25
5 -_ 2
Order each set of numbers from least to greatest. 1 √ _ 2 √ √ 30. 5_ , 2.1, 4 , 6 31. 4. 23, 4 _ , 18 , 16 5
4
3
Order each set of numbers from greatest to least. 1 32. -10, -10 _ , -1.05, - √ 105 2
1 17 33. - √ 14 , -4 _ , -_ , -3.8 10
4
ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 34. a2 = 49
35. 300 = h2
36. y2 = 22
37. 0.0058 = k2
38. 5p2 = 315
39. 2d2 = 162
40. 190.5 = 1.5b2
41. 0.1x2 = 0.169
472 Chapter 9 Real Numbers and Right Triangles
42. TRACK AND FIELD Use the information at the left. Suppose American Stacy Dragila reached a winning height of about 15 feet in the 2000 Olympics. About how fast was she running? Round to the nearest tenth. 43. PHYSICS The formula h = 16t2 measures the time t in seconds that it takes for an object to fall from a height of h feet and hit the ground. How long would it take a marble to hit the ground if it was dropped off a cliff with a height of 150 feet? Round to the nearest tenth. Determine whether each statement is sometimes, always, or never true. 44. A whole number is an integer. 45. An irrational number is a negative integer. 46. A repeating decimal is a real number. 47. An integer is a whole number. 48. FLOORING A square room has an area of 324 square feet. The homeowners plan to cover the floor with 6-inch square tiles. How many tiles will be in each row on the floor? Real-World Link To find the height h in feet that a pole vaulter can reach, coaches can v2 use the formula h = 64 where v is the velocity of the pole vaulter in feet per second. Source: American Institute of Physics
Give a counterexample for each statement. 49. All square roots are irrational numbers. 50. All rational numbers are integers. 152 ? 51. What is the value of x to the nearest tenth if x2 - 42 = √ 52. GEOMETRY Use the formula for the area of a circle A = r2, where A represents the area, r represents the radius, and is approximately equal to 3.14, to find the radius of the circle with an area of 28.26 square inches.
R
30fd , where s represents the speed of a car in miles 53. CARS The formula s = √ per hour, d represents the distance the car skidded in feet, and f is friction, can be used to determine how fast a car was traveling before it skidded to a stop. The table shows some different values of f. At an accident scene, a car made 100-foot skid marks before hitting another car. If the speed limit was 55 miles per hour, was the car speeding before applying the brakes on a dry, concrete road? Explain.
4YPE OF 3URFACE
EXTRA
PRACTICE
2OAD #ONDITIONS
#ONCRETE
!SPHALT
7ET
$RY
See pages 781, 802. Self-Check Quiz at pre-alg.com
H.O.T. Problems
54. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would solve equations by finding square roots. 55. OPEN ENDED Give an example of a number that is an integer and a rational number. 56. CHALLENGE Tell whether the product of a rational number like 8 and an irrational number like 0.101001000… is rational or irrational. Explain your reasoning. Lesson 9-2 The Real Number System
Matthew Stockman/Getty Images
473
57. Which One Doesn’t Belong? Identify the number that does not belong with the other three. Explain your reasoning. 50 -_ 2
50.1
58.
√ 50
-50.1
Writing in Math Explain the relationship between the area of a square and the length of its sides. Give an example of a square whose side length is irrational and an example of a square whose side length is rational.
60. Which number can only be classified as a rational number?
59. The time t in seconds it takes an object to fall d feet can be estimated by using d = 0.5(32)t2. If a ball is dropped from the top of a 120-foot building, how long does it take to hit the ground? A 1.9 s
C 3.8 s
B 2.7 s
D 7.5 s
F -2
1 G _
H √ 2
2
1 < 61. For what value of x is _ √x true?
A -2
1 B _
1 C _ 2
4
J 2 √x
<x
D 2
Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 62. √ 54 63. - √ 126 64. √ 8.67 65. - √ 19.85 66. Solve 9 - 2d ≤ 23 and check your solution. Graph the solution on a number line. (Lesson 8-8) WEATHER The table shows the heat index and relative humidity for an air temperature of 75°F. (Lesson 7-8) Relative Humidity Heat Index (°F)
0% 69
5% 69
10% 70
15% 71
20% 72
25% 72
30% 73
35% 73
40% 74
45% 74
50% 75
67. Make a scatter plot and draw a line of fit. 68. Use the line of fit to predict the heat index for a relative humidity of 70%. Express each ratio as a unit rate. Round to the nearest hundredth, if necessary. (Lesson 6-1) 69. $8 for 15 cupcakes
70. 120 miles on 4.3 gallons
71. 3 feet of snow in 5 hours
72. $22 in 5 hours
PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 73. 18 + 57 + x = 180
74. x + 27 + 54 = 180
75. 85 + x + 24 = 180
76. x + x + x = 180
77. 2x + 3x + 4x = 180
78. 2x + 3x + 5x = 180
474 Chapter 9 Real Numbers and Right Triangles
Learning Geometry Vocabulary Many of the words used in geometry are commonly used in everyday language. For example, the photo shows rays of light. The everyday meanings of ray can be used to better understand its mathematical meaning. The table below shows the meanings of some geometry terms you will use throughout this chapter. Term
ray
degree
Everyday Meaning
Mathematical Meaning
any of the thin lines, or beams, of light that appear to come from a bright source • a ray of light
a part of a line that extends from a point indefinitely in one direction
extent, amount, or relative intensity • third degree burns
a common unit of measure for angles
characterized by sharpness or severity • an acute pain
an angle with a measure that is greater than 0° and less than 90°
not producing a sharp impression • an obtuse statement
an angle with a measure that is greater than 90° but less than 180°
acute
obtuse
Source: Merriam Webster’s Collegiate Dictionary
Reading to Learn 1. Write a sentence using each term listed above. Be sure to use the everyday meaning of the term. 2. RESEARCH Use the Internet or a dictionary to find the everyday meaning of each term listed below. Compare them to their mathematical meaning. Note any similarities and/or differences. a. midpoint
b. converse
c. indirect
3. RESEARCH Use the Internet or dictionary to determine which of the following words are used only in mathematics. vertex
equilateral
similar
scalene
side
isosceles
Reading Math Learning Geometry Vocabulary Digital Vision/PunchStock
475
9-3
Triangles Animation pre alg.com
Main Ideas • Find the missing angle measure of a triangle.
There is a relationship among the measures of the angles of a triangle.
• Classify triangles by properties and attributes.
New Vocabulary line segment triangle vertex acute angle right angle obtuse angle straight angle acute triangle obtuse triangle right triangle congruent scalene triangle isosceles triangle equilateral triangle
Step 1
Use a straightedge to draw a triangle on a piece of paper. Then cut out the triangle and label the vertices X, Y, and Z.
Step 2
Fold the triangle as shown so that point Z lies on side XY as shown. Label the back of ∠Z as ∠2.
Step 3
Fold again so point X meets the vertex of ∠2. Label the back of ∠X as ∠1.
Step 4
Fold so point Y meets the vertex of ∠2. Label the back of ∠Y as ∠3.
Z X
Z
Z Y
X
2
X1 2
Y
Step 2
Step 1
Z Y
X1 2 3 Y
Step 3
Step 4
MAKE A CONJECTURE What is the sum of the measures of ∠1, ∠2, and ∠3? Explain your reasoning.
Angle Measures of a Triangle A line segment is part of a line containing two endpoints and all of the points between them. A triangle is formed by three line segments that intersect only at their endpoints. Each pair of segments forms an angle of the triangle.
The vertex of each angle is a vertex of the triangle. Vertices is the plural of vertex.
Reading Math Line Segment The symbol for line . segment XY is XY
Triangles are named by the letters at their vertices. Triangle XYZ, written XYZ, is shown. vertex
X
side
angle Y
Z
The sides are XY, YZ, and XZ. The vertices are X, Y, and Z. The angles are ∠ X, ∠Y, and ∠ Z.
The activity above suggests a relationship about the angles of any triangle. Angles of a Triangle Words The sum of the measures of the angles of a triangle is 180°. Model
Symbols
x˚ y˚
476 Chapter 9 Real Numbers and Right Triangles
z˚
x + y + z = 180
EXAMPLE
Find Angle Measures
Find the value of x in ABC.
A
m∠A + m∠B + m∠C = 180
x˚
x + 58 + 55 = 180 x + 113 = 180 x + 113 - 113 = 180 - 113
B
58˚
55˚
C
x = 67
1. Find the value of ∠E in DEF if m∠D = 62° and m∠F = 39°.
EXAMPLE
Use Ratios to Find Angle Measures
ALGEBRA The measures of the angles of a certain triangle are in the ratio 1:4:7. What are the measures of the angles? Words
The sum of the measures is 180°.
Variables
Let x represent the measure of the first angle, and 7x the measure of the third angle.
Equation
x + 4x + 7x = 180
the measure of a second angle,
x + 4x + 7x = 180 Write the equation. 12x = 180 Combine like terms. 180 12x _ =_
Check for Accuracy 15 + 60 + 105 = 180. So, the answer is correct.
12
12
x = 15
Divide each side by 12. Simplify.
Since x = 15, 4x = 4(15) or 60, and 7x = 7(15) or 105. The measures of the angles are 15°, 60°, and 105°.
2. The measures of the angles of a certain triangle are in the ratio 1:3:6. What are the measures of the angles? Personal Tutor at pre-alg.com
Classify Triangles Angles can be classified by their degree measure. Types of Angles Acute Angle
Right Angle
Obtuse Angle
Straight Angle
This symbol is used to indicate a right angle.
A
0° < m∠A < 90°
A
A
m∠ A = 90°
A
90° < m∠ A < 180°
m∠ A = 180°
Lesson 9-3 Triangles
477
EXAMPLE
Classify Angles
Classify each angle as acute, obtuse, right, or straight. a.
b.
A B
c. D
G 40˚
125˚
J
F C
E
m∠ABC > 90° So, ∠ABC is obtuse.
3A. 154°
H
m∠DEF = 90° So, ∠DEF is right.
m∠GHJ < 90° So, ∠GHJ is acute.
3B. 88°
3C. 180°
Triangles can be classified by their angles and their sides. Congruent sides have the same length.
Classify Triangles Acute Triangle
Obtuse Triangle
40˚ 80˚ 60˚
Reading Math Congruent Segments Tick marks on the sides of a triangle indicate that those sides are congruent.
Right Triangle
30˚ 110˚40
45˚
˚
45˚
all acute angles
one obtuse angle
one right angle
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
at least two sides congruent
all sides congruent
no congruent sides
EXAMPLE
Classify Triangles
Classify the triangle by its angles and by its sides. R
T
45˚
Angles: RST has a right angle. 45˚
S
Sides: RST has two congruent sides.
So, RST is a right isosceles triangle.
Classify each triangle by its angles and by its sides. 4A. 4B. Ón Ón xn £Ó{ {Ó nä 478 Chapter 9 Real Numbers and Right Triangles
Extra Examples at pre-alg.com
Example 1 (p. 477)
Find the value of x in each triangle. Then classify each triangle as acute, right, or obtuse. 1.
2.
83˚ 72˚
Example 2 (p. 477)
Example 3 (p. 478)
Example 4 (p. 478)
3. 61˚
x˚
x˚
27˚
48˚ x˚
29˚
4. ALGEBRA Triangle EFG has angles whose measures are in the ratio 1:5:9. What are the measures of the angles? Classify each angle as acute, obtuse, right, or straight. 5. 55°
6. 140°
Classify each indicated triangle by its angles and by its sides. 8.
7.
Lima
90˚ 110˚
45˚ 45˚
45˚
9.
Youngstown
Ohio
25˚
75˚
Cincinnati
75˚
30˚
HOMEWORK
HELP
For See Exercises Examples 10–15 1 16–17 2 18–28 3 29–34 4
Find the value of x in each triangle. Then classify each triangle as acute, right, or obtuse. 10.
11.
x˚
12. 57˚
32˚
63˚ 68˚
13.
x˚
15. 28˚
33˚
36˚
x˚
14. 45˚
x˚
x˚
x˚ 43˚ 71˚
62˚
16. ALGEBRA The measures of the angles of a triangle are in the ratio 1:3:5. What is the measure of each angle? 17. ALGEBRA Determine the measures of the angles of ABC if the measures of the angles are in the ratio 1:1:16. Classify each angle as acute, obtuse, right, or straight. 18. 40°
19. 70°
20. 65°
21. 85°
22. 95°
23. 110°
24. 155°
25. 140°
26. 38°
27. 127°
28. TIME What type of angle is formed by the hands on a clock at 6:00? Lesson 9-3 Triangles
479
Classify each indicated triangle by its angles and by its sides. 29.
30.
60˚
31. 60˚
40˚
40˚
30˚ 60˚
60˚
32.
33.
ÃÜÀÌ
34.
D[XhWiaW
40˚
V
35˚
V
70˚ 70˚
110˚
35˚
Sketch each triangle. If it is not possible to sketch the triangle, write not possible. Real-World Link Tony Hawk was the first skateboarder to perform the 900 during the X-Games. He rode off a ramp, spun 900° in mid-air, and made a perfect landing, all on a skateboard.
35. acute scalene 37. right equilateral
36. obtuse and not scalene 38. obtuse equilateral
BASEBALL If you swing the bat too early or too late, the ball will probably go foul. The best way to hit the ball is at a right angle. Classify each angle shown. 39.
J
40.
K
R
S
41.
Source: skateboardlink.com
Incoming Path Angle 90˚
D
O
Angle 90˚
T
Angle 90˚
L
T
42. SKATEBOARDING Refer to the information at the left. How many revolutions did Tony make performing the 900? EXTRA
PRACTICE
See pages 782, 802. Self-Check Quiz at pre-alg.com
ALGEBRA Find the measures of the angles in each triangle. 43.
44.
5x ˚ 3x ˚
x˚
(x 5)˚
x˚
45. 2x ˚
85˚
(2x 15)˚ 7x ˚
H.O.T. Problems
46. OPEN ENDED Draw an obtuse isosceles triangle to represent a real-world object. Would the object still be useful if the triangle were acute? Explain. 47. CHALLENGE Numbers that can be represented by a triangular arrangement of dots are called triangular numbers. The first three triangular numbers are 1, 3, and 6. Find the next three triangular numbers. 48. SELECT A TOOL The measure of ∠1 is twice the measure of ∠2. The measure of ∠3 is 40° less than the measure of ∠2. Which of the following tools would you use to determine the measures of the three angles? Justify your selection(s). Then use the tool(s) to solve the problem. draw a model
480 Chapter 9 Real Numbers and Right Triangles Jamie McDonald/Getty Images
paper/pencil
calculator
49.
Writing in Math How do the angles of a triangle relate to each other? Include drawings of two triangles with their angles labeled.
50. The measures of the angles of a brick paver are 30°, 90°, and 60°. Which triangle most likely has these angle measures? A
51. A long piece of paper is folded so that the lower edge of the strip forms a right angle with itself. Classify ∠3.
C
B
D
F acute
H right
G obtuse
J
straight
ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. (Lesson 9-2) 52. m2 = 81
53. 196 = y2
x2 55. _ = 51
54. 168 = 2p2
2
Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 56. - √ 5.25
57. - √ 17.3
59. √ 140.57
58. √ 38.75
60. Twenty-six is 25% of what number? (Lesson 6-5)
{
Ài} Ì Ã v iÌÀV Ìî
Î
ΰΠӰ£ä
£°nÎ
£°È{
£°ÈÎ
£°ÈÓ
Ó £
i Õ
ÃÛ
iÜ
9 À
>
®
ä i « à V À> }i à } i iÃ
62. WEATHER The time a storm will hit an area can be predicted using the formula d ÷ s = t where d is the distance in miles an area is from the storm, s is the speed in miles per hour of the storm, and t is the travel time in hours of the storm. Suppose it is 11:00 A.M. and a storm is heading toward a town at a speed of 30 miles per hour. The storm is about 150 miles from the town. At what time will the storm hit? (Lesson 3-8)
Ûi ½i "ÕÌt
61. AIRPORTS The graph shows North America’s busiest cargo airports in 2003. What is the difference in cargo handling between Anchorage and New York? (Lesson 5-8)
À«ÀÌ -ÕÀVi\ À«ÀÌÃ ÕV ÌiÀ>Ì>
PREREQUISITE SKILL Find the value of each expression. (Lesson 4-1) 63. 122
64. 152
65. 182
66. 212
67. 242
68. 272
Lesson 9-3 Triangles
481
CH
APTER
9
Mid-Chapter Quiz Lessons 9-1 through 9-3 2
Find each square root. (Lesson 9-1) 1. √ 36 3. ± √ 81
2. - √ 169 2 4. √m
8t 21. CLOCKS The formula L = _ represents π2 the swing of a pendulum, where L is the length of the pendulum in feet and t is the time in seconds that it takes to swing back and forth. How long does it take a 4-foot pendulum to swing back and forth?
Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson 9-1) 5. √ 51 7. √ 17
6. √ 88 8. √ 111
(Lesson 9-2)
Classify each angle measure as acute, obtuse, right, or straight. (Lesson 9-3)
9. MULTIPLE CHOICE Which statement is NOT true? (Lesson 9-1)
22. 83°
23. 180°
24. 115°
25. 90°
Classify each triangle by its angles and by its sides. (Lesson 9-3)
A 6 < √ 39 < 7 < 10 B 9 < √89
26.
C -7 > - √ 56 > -8 D -4 < - √ 17 < -5
xÎ {Ó
{Ó
10. GARDENING Marisa wants to put a fence around her square vegetable garden that A = s, has an area of 169 square feet. If √ where s is the length of one side and A is the area, how many feet of fencing does she need to enclose the garden?
27.
È
28.
ÎÇ
29. Èäc Îäc Èäc
Èäc Èäc
(Lesson 9-1)
Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. (Lesson 9-2) 11. 0.3 − 13. 15.1
12. - √ 49 56 14. _ 8
30. MULTIPLE CHOICE Refer to the figure shown. Bob lives in Salsburg, does his grocery shopping in Richmond, and sees a doctor in Thornville. What is the measure of the angle formed when Bob travels from Salsburg to Richmond and then to Thornville? (Lesson 9-3) 2ICHMOND
ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. (Lesson 9-2) 15.
m2
= 68
16. 131 =
n2
3ALSBURG
££äc
ÎÓc
17. 600 = 1.5x2
18. 2b2 = 98
F 28°
H 48°
19. 259.2 = 5y2
20. 3.6r2 = 518.4
G 38°
J 180°
482 Chapter 9 Real Numbers and Right Triangles
4HORNVILLE
EXPLORE
9-4
Algebra Lab
The Pythagorean Theorem ACTIVITY 1 Dot paper can be used to find the area of certain geometric figures. Consider the following examples. Find the area of each shaded region if each square square unit.
1 1 A=_ (1) or _ unit2 2
2
1 A=_ (2) or 1 unit2 2
represents one
1 A=_ (4) or 2 units2 2
The area of other figures can be found by first separating the figure into smaller regions and then finding the sum of the areas of the smaller regions.
A = 2 units2
A = 5 units2
A = 4 units2
EXERCISES Find the area of each figure. 1.
2.
3.
4.
Explore 9-4 Algebra Lab: The Pythagorean Theorem
483
ACTIVITY 2 Let’s investigate the relationship that exists among the sides of a right triangle. In each diagram shown, notice how a square is attached to each side of a right triangle. Triangle 1
Triangle 2
Triangle 3
Square C
Square C
Square C Square A
Square A
Square A Square B
Square B
Square B
Triangle 4
Triangle 5
3QUARE #
Square C Square A
3QUARE ! 3QUARE " Square B
Copy the table. Then find the area of each square that is attached to the triangle. Record the results in your table. Triangle
Area of Square A (units2)
Area of Square B (units2)
Area of Square C (units2)
1 2 3 4 5
EXERCISES 5. Refer to your table. How does the sum of the areas of square A and square B compare to the area of square C? 6. Draw a right triangle on centimeter grid paper. Count to find the measures of the legs and use the relationship you discovered to calculate the measure of the hypotenuse. Measure to verify your answer. 7. Refer to the diagram at the right. If the lengths of the sides of a right triangle are whole numbers such that a2 + b2 = c2, the numbers a, b, and c are called a Pythagorean Triple. Tell whether each set of numbers is a Pythagorean Triple. Explain. a. 3, 4, 5
b. 5, 7, 9
484 Chapter 9 Real Numbers and Right Triangles
c. 6, 9, 12
d. 7, 24, 25
a
c
b
9-4
The Pythagorean Theorem Interactive Lab pre-alg.com
Main Ideas • Use the Pythagorean Theorem to find the length of a side of a right triangle.
In the diagram, three squares with sides 3, 4, and 5 units are used to form a right triangle.
• Use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle.
b. What relationship exists among the areas of the squares?
New Vocabulary legs hypotenuse Pythagorean Theorem solving a right triangle converse
5 units 3 units
a. Find the area of each square.
c. Draw three squares with sides 5, 12, and 13 units so that they form a right triangle. What relationship exists among the areas of these squares?
4 units
The Pythagorean Theorem In a right triangle,
hypotenuse
the sides that are adjacent to the right angle are called the legs. The side opposite the right angle is the hypotenuse.
legs
The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse for any right triangle. Pythagorean Theorem If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Words
Model c
a
c2 = a2 + b2
Example
52 = 32 + 42 25 = 9 + 16 25 = 25
b
Reading Math Hypotenuse The hypotenuse is the longest side of a right triangle.
Symbols
EXAMPLE
Find the Length of the Hypotenuse
Find the length of the hypotenuse of the right triangle. c2 = a2 + b2
Pythagorean Theorem
c2
Replace a with 12 and b with 16.
=
122
+
162
c2
= 144 + 256 Evaluate
c2
= 400
√ c2 = √ 400 c = 20
122
and
162.
Add 144 and 256.
12 ft
c ft
16 ft
Take the positive square root of each side. The length of the hypotenuse is 20 feet.
1. Find the length of the hypotenuse of a right triangle if its legs are 24 inches and 45 inches long. Lesson 9-4 The Pythagorean Theorem
485
If you know the lengths of two sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side. This is called solving a right triangle.
EXAMPLE Square Roots In Lesson 9-1, you learned that a number has both a positive and negative square root. Since we are finding the lengths of sides of triangles, we calculate the positive square root.
Solve a Right Triangle
Find the length of the leg of the right triangle. c2 = a2 + b2
14 cm
Pythagorean Theorem a cm
142 = a2 + 102
Replace c with 14 and b with 10.
196 = a2 + 100
Evaluate 142 and 102.
196 - 100 =
a2
+ 100 - 100 Subtract 100 from each side.
96 = a2 √ 96 =
10 cm
Simplify.
√ a2
Take the square root of each side.
2nd [ √ ] 96 ENTER 9.797958971
The length of the leg is about 9.8 centimeters. {£ V
X V
2. Find the length of the leg of the right triangle.
{ä V
GRIDDABLE A painter positions a 20-foot ladder against a house so that the base of the ladder is 4 feet from the house. About how many feet does the ladder reach on the side of the house? Round to the nearest tenth.
20 ft 4 ft
Read the Test Item The ladder, ground, and side of the house form a right triangle. You know the hypotenuse and one leg of a right triangle. You need to find the other leg. Solve the Test Item Use the Pythagorean Theorem to find how high the ladder reaches on the side of the house. c2 = a2 + b2 202 = 42 + b2 400 = 16 +
b2
Pythagorean Theorem Replace c with 20 and a with 4. Evaluate 202 and 42.
400 - 16 = 16 + b2 - 16 Subtract 16 from each side. 384 = b2 √ 384 = √ b2
19.6 ≈ b
Simplify. Take the square root of each side. Round to the nearest tenth.
The ladder reaches about 19.6 feet on the side of the house. 486 Chapter 9 Real Numbers and Right Triangles
Extra Examples at pre-alg.com
Fill in the Answer Grid
3. GRIDDABLE A doorway is 2.7 feet wide and 8.4 feet high. What is the longest piece of drywall in feet that can be taken through this doorway? Round to the nearest tenth. Personal Tutor at pre-alg.com
Converse of the Pythagorean Theorem The Pythagorean Theorem is written in if-then form. If you reverse the statements after if and then, you have formed the converse of the Pythagorean Theorem.
Pythagorean Theorem If a triangle is a right triangle, then c2 = a2 + b2.
Since there is no place for a negative sign or a fraction on the grid, griddable answers are never negative numbers or fractions.
Write 19.6 in the answer boxes and write only one digit in each answer box. Fill in one bubble for every answer box that you have written in. Be sure not to fill in a bubble under a blank answer box.
If c2 = a2 + b2, then a triangle is a right triangle.
Converse
The converse of the Pythagorean Theorem is also true. You can use the converse to determine whether a triangle is a right triangle.
EXAMPLE
Identify a Right Triangle
The measures of three sides of a triangle are given. Determine whether each triangle is a right triangle. a. 9 m, 12 m, 15 m c2 = a2 + b2 152 92 + 122 225 81 + 144 225 = 225
Pythagorean Theorem Replace c with 15, a with 9, and b with 12. Evaluate 152, 92, and 122. Simplify.
The triangle is a right triangle. b. 6 in., 7 in., 12 in. c2 = a2 + b2 122 62 + 72 144 36 + 49 144 ≠ 85
Pythagorean Theorem Replace c with 12, a with 6, and b with 7. Evaluate 122, 62, and 72. Simplify.
The triangle is not a right triangle.
4A. 8 in., 9 in., 12 in.
4B. 15 mm, 20 mm, 25 mm Lesson 9-4 The Pythagorean Theorem
487
Example 1 (p. 485)
Find the length of the hypotenuse in each right triangle. Round to the nearest tenth, if necessary. 1.
2.
12 ft
6 ft
cm
15 m
c ft 20 m
Example 2 (p. 486)
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 3. a = 8, b = ?, c = 17
Example 3 (pp. 486–487)
Example 4 (p. 487)
4. a = ?, b = 24, c = 25
5. GRIDDABLE Kendra is flying a kite. The length of the kite string is 55 feet, and she is positioned 40 feet away from the point directly beneath the kite. About how high is the kite in feet? The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 6. 5 cm, 7 cm, 8 cm
HOMEWORK
HELP
For See Exercises Examples 8–13 1 14–19 2 20–25 4 46–47 3
7. 20 ft, 48 ft, 52 ft
Find the length of the hypotenuse in each right triangle. Round to the nearest tenth, if necessary. 8.
9.
10.
24 in.
cm
6m
24 m 10 in.
cm
c in. 45 m
8m
11.
12.
13.
7.2 cm 2.7 cm
c ft
40 ft
12.8 m
cm
c cm 13.9 m
30 ft
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 14. a = 30, b = ?, c = 50
15. a = ?, b = 35, c = 37
16. a = ?, b = 12, c = 19
17. a = 7, b = ?, c = 14
Find each missing measure to the nearest tenth. 18.
19. 45 ft
17 m
28 m
x ft
30 ft
488 Chapter 9 Real Numbers and Right Triangles
xm
The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. 20. a = 5, b = 8, c = 9
21. a = 16, b = 30, c = 34
22. a = 18, b = 24, c = 30
23. a = 24, b = 28, c = 32
24. a = √ 21 , b = 6, c = √ 57
25. a = 11, b = √ 55 , c = √ 177
26. GYMNASTICS The floor exercise mat measures 40 feet by 40 feet. Find the measure of the diagonal. 27. TELEVISION The size of a flat-screen television is determined by the length of the diagonal of the screen. If a 35-inch television screen is 26 inches long, what is its height to the nearest inch? Real-World Link The floor exercise mat is a square of plywood covered by a 2-inch padding and mounted on 4” springs. Source: The Gymnastics Place!
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 28. a = 8.1, b = 3.5, c = ?
29. a = 10.4, b = 16.9, c = ?
30. a = 27, b = ?, c = 61
31. a = ?, b = 73, c = 82
123 =, c = 22 32. a = ?, b = √
33. a = √ 177 =, b = ?, c = 31
34. GEOMETRY If the vertex of an angle lies on a circle, the angle is called an inscribed angle. All angles inscribed in a semicircle are right angles. In the figure, ∠ACB is an inscribed AB is 17 and the right angle. If the measure of AC is 8, find the measure of BC. measure of
C 8
A
B
17
ANALYZE GRAPHS Find the length of the hypotenuse. Write your answer as a square root. 35.
36.
Y
37.
Y
!
Y
2 &
$ "
X
X
"
#
X
"
" 4
3
'
TRAVEL Europe’s largest town square is the Rynek Glowny located in Krakow, Poland. It covers approximately 48,400 square yards. 38. How many feet long is a side of the square? 39. To the nearest foot, approximately what is the diagonal distance across Rynek Glowny?
EXTRA
PRACTICE
See pages 782, 802. Self-Check Quiz at pre-alg.com
B
ART For Exercises 40 and 41, use the plasterwork design shown. 40. If the sides of the square measure 6 inches, AB ? what is the length of AB 41. What is the perimeter of the design if 128 inches? measures √
A Lesson 9-4 The Pythagorean Theorem
Steven E. Sutton/DUOMO
489
H.O.T. Problems
42. OPEN ENDED State three numbers that could be the measures of the sides of a right triangle. Justify your answer. 43. FIND THE ERROR Marcus and Allyson are finding the missing measure of the right triangle shown. Who is correct? Explain your reasoning. Marcus c2 = a2 + b2 152 = 92 + b2 12 = b
Allyson c2 = a 2 + b 2 c2 = 92 + 152 c2 ≈ 125
15 ft
9 ft
c ft
44. CHALLENGE The hypotenuse of an isosceles right triangle is 8 inches. Is there enough information to find the length of the legs? If so, find the length of the legs. If not, explain why not. 45.
Writing in Math
How do the lengths of the sides of a right triangle relate to each other? Include an example of a set of numbers that represents the measures of the legs and hypotenuse of a right triangle.
46. Find the amount of edging needed to enclose the triangular flower bed. A 10 yd B 16 yd
YD
C 18 yd
47. GRIDDABLE How far is it in feet from home plate to second base? Round to the nearest tenth.
Ó` >Ãi Èä vÌ £ÃÌ >Ãi
ÎÀ` >Ãi Èä vÌ
YD
D 24 yd
i *>Ìi
Find the value of x in each triangle. Then classify each triangle as acute, right, or obtuse. (Lesson 9-3) 48.
49.
x˚ 46˚
33˚
50.
63˚ x˚
27˚
x˚ 48˚
54˚
Name all of the sets of numbers to which each real number belongs. Let N = natural numbers, W = whole numbers, Z = integers, Q = rational numbers, and I = irrational numbers. (Lesson 9-2)
51. -5
52. 0. 4
53. 63
54. 7.4
55. GRADES Tobias’ average for five quizzes is 86. If he wants to have an average of at least 88 for six quizzes, what is the lowest score he can receive on his sixth quiz? (Lesson 5-9)
PREREQUISITE SKILL Simplify each expression (Lessons 1-2 and 4-2) 56. (2 + 6)2 + (-5 + 6)2
57. (-4 + 3)2 + (0 - 2)2
490 Chapter 9 Real Numbers and Right Triangles
58. [3 + (-1)]2 + (8 - 4)2
EXTEND
Algebra Lab
9-4
Graphing Irrational Numbers
In Lesson 2-1, you learned to graph integers on a number line. Irrational numbers can also 53 . To graph √ 53 , construct be graphed on a number line. Consider the irrational number √ √ a right triangle whose hypotenuse measures 53 units.
ACTIVITY Step 1 Find two numbers whose squares have a sum of 53. Since 53 = 49 + 4 or 72 + 22, one pair that will work is 7 and 2. These numbers will be the lengths of the legs of the right triangle. Step 2 Draw the right triangle • First, draw a number line on grid paper.
0 1 2 3 4 5 6 7 8
• Next, draw a right triangle whose legs measure 7 units and 2 units. Notice that this triangle can be drawn in two ways. Either way is correct.
7 units 2 units 01 2 3 4 5 6 7 8
01 2 3 4 5 6 7 8
53 . Step 3 Graph √ • Open your compass to the length of the hypotenuse. • With the tip of the compass at 0, draw an arc that intersects the number line at point B. 53 • The distance from 0 to B is √ 53 ≈ 7.3. units. From the graph, √
01 2 3 4 5 6 7 8
B 01 2 3 4 5 6 7 8
ANALYZE THE RESULTS Use a compass and grid paper to graph each irrational number on a number line. 1. √ 5
3. √ 45
2. √ 20
4.
√ 97
. 5. Describe two different ways to graph √34 6. Explain how the graph of √2 can be used to locate the graph of √3. Extend 9-4 Algebra Lab: Graphing Irrational Numbers
491
9-5
The Distance Formula
Main Idea • Use the Distance Formula to determine lengths on a coordinate plane.
New Vocabulary Distance Formula
The graph of points N(3, 0) and M(-4, 3) is shown. A horizontal segment is drawn from M, and a vertical segment is drawn from N. The intersection is labeled P.
y
M (⫺4, 3)
P
O
a. Name the coordinates of P.
N (3, 0) x
b. Find the distance between M and P. c. Find the distance between N and P. d. Classify MNP. e. What theorem can be used to find the distance between M and N? f. Find the distance between M and N.
The Distance Formula A line segment is a part of a line that contains two endpoints and all of the points between the endpoints. y
A line segment is named by its endpoints.
M
O
N x
The segment can be written as MN or NM.
To find the length of a segment on a coordinate plane, you can first extend horizontal and vertical segments from the vertices to form a right triangle. Then use the Pythagorean Theorem to find the length of the segment. You can also use the Distance Formula, which is based on the Pythagorean Theorem. Distance Formula Words Look Back To review the notation (x1 , y1) and (x2 , y2 ), see Lesson 7-5.
The distance d between two points with coordinates (x1, y1) and (x2, y2), is given by d = √(x - x ) 2 + (y - y ) 2 .
492 Chapter 9 Real Numbers and Right Triangles
2
1
2
1
y
Model ( x 1, y 1)
( x 2, y 2)
d
O
x
EXAMPLE
Use the Distance Formula
Find the distance between G(-3, 1) and H(2, -4). Round to the nearest tenth, if necessary. G (⫺3, 1)
Use the Distance Formula. Substitution You can use either point as (x1, y1). The distance will be the same.
(x2 - x1)2 + (y2 - y1)2 √
Distance Formula
GH =
[2 - (-3)]2 + (-4 - 1)2 √
(x1, y1) = (-3, 1), (x2, y2) = (2, -4)
GH =
(5)2 + (-5)2 √
Simplify.
d=
y
GH = √ 25 + 25
Evaluate 52 and (-5)2.
GH = √ 50
Add 25 and 25.
GH ≈ 7.1
Take the square root.
x
O
H (2, ⫺4)
1. Find the distance between A(5, -6) and B(1, 2). Round to the nearest tenth, if necessary.
Reading Math Segment Measure The symbol GH means the measure of segment GH.
EXAMPLE
Use the Distance Formula to Solve a Problem
GEOMETRY Find the perimeter of ABC to the nearest tenth.
y
A (⫺2, 3)
First, use the Distance Formula to find the length of each side of the triangle. −− Side AB: A(-2, 3), B(2, 2) d=
B (2, 2)
O x
(x2 - x1)2 + (y2 - y1)2 √
C (0, ⫺3)
[2 - (-2)]2 + (2 - 3)2 √ AB = √ (4)2 + (-1)2 AB =
16 + 1 or √ 17 AB = √ −− Side BC : B(2, 2), C(0, -3) d=
(x2 - x1)2 + (y2 - y1)2 √
−− Side CA: C(0, -3), A(-2, 3) d=
(x2 - x1)2 + (y2 - y1)2 √
BC =
(0 - 2)2 + (-3 - 2)2 √ BC = √ (-2)2 + (-5)2
(-2 - 0)2 + [3 - (-3)]2 √ CA = √ (-2)2 + (6)2
Common Misconception
4 + 25 or √ 29 BC = √
CA = √ 4 + 36 or √ 40
To find the sum of square roots, do not add the numbers inside the square root symbols. √ 17 + √ 29 + √ 40 ≠ √ 86
Then add the lengths of the sides to find the perimeter.
CA =
√ 17 + √ 29 + √ 40 ≈ 4.123 + 5.385 + 6.325 ≈ 15.833
The perimeter is about 15.8 units.
2. Find the perimeter of XYZ with vertices X(1, 3), Y(3, -4), and Z(-4, 1) to the nearest tenth. Extra Examples at pre-alg.com
Lesson 9-5 The Distance Formula
493
TRAVEL The Yeager family is visiting Washington, D.C. A unit on the coordinate system of their map shown at the right is 0.05 mile. Find the distance between the Department of Defense at (-2, 9) and the Madison Building at (3, -3). (x2 - x1)2 + (y2 - y1)2 √
Benjamin Banneker helped to survey and lay out Washington, D.C. He also made all the astronomical and tide calculations for the almanac he published. Source: World Book
6 4
U.S. Capitol
[3 - (-2)]2 + (-3 - 9)2 √ d = √ (5)2 + (-12)2
Add.
d = √ 25 + 144
Simplify.
d = √ 169 or 13
Take the square root.
d=
Real-World Link
8
2
Use the distance formula. d=
Department of Defense
–2
Madison Building
–4 –4
–2
2
4
6
The distance between the two buildings is 13 units on the map. Since each unit equals 0.05 mile, the distance between the two buildings is 0.05 · 13 or 0.65 mile.
3. TRAVEL Find the distance between the Madison Building at (3, -3) and the U.S. Capitol at (0, 0). Personal Tutor at pre-alg.com
Example 1 (p. 493)
Find the distance between each pair of points. Round to the nearest tenth, if necessary. 1. A(-1, 3), B(8, -6)
Example 2 (p. 493)
Example 3 (p. 494)
HOMEWORK
HELP
For See Exercises Examples 5–10 1 11, 12 2 13, 14 3
3. GEOMETRY Triangle EFG has vertices E(1, 4), F(-3, 0), and G(4, -1). Find the perimeter of EFG to the nearest tenth. 4. ARCHAEOLOGY An archaeologist creates a coordinate system to record where artifacts were discovered. A unit on the grid represents 5 feet. Find the distance between two artifacts if one artifact was found at (-3, 1) and the other was found at (-6, -5) on the grid. Round to the nearest tenth.
Find the distance between each pair of points. Round to the nearest tenth, if necessary. 5. J(5, -4), K(-1, 3)
6. C(-7, 2), D(6, -4)
7. E(-1, -2), F(9, -4)
8. V(8, -5), W(-3, -5)
9. S(-9, 0), T(6, -7)
494 Chapter 9 Real Numbers and Right Triangles The Granger Collection, NY
2. M(4, -2), N(-6, -7)
10. M(0, 0), N(-7, -8)
GEOMETRY Find the perimeter of each figure. 11.
12.
y
X (⫺2, 3)
y A (4, 4)
Y (3, 0)
O
x
O
x
C (⫺2, ⫺2) B (1, ⫺4)
Z (⫺2, ⫺4)
13. LANDSCAPING Len set up a coordinate system with units of feet to locate the positions of his flowers. He planted hostas at (1, 5) and a rose bush at (-6, 3). How far apart are the two plants? Round to the nearest tenth of a foot. Real-World Link The first U.S. public zoo was established in Philadelphia in 1874. Source: philadelphiazoo.org
14. ZOO Beth is looking at a map of the zoo that is laid out on a coordinate system. Beth is at (1, -1). The gorilla house is at (-2, -4) and the reptile exhibit is at (3, 2). Is Beth closer to the gorilla house or the reptile exhibit? 15. GEOMETRY Determine whether MNP with vertices M(3, -1), N(-3, 2), and P(6, 5) is isosceles. Explain your reasoning. Find the distance between each pair of points. Round to the nearest tenth, if necessary.
(
) (
)
16. Q 51, 3 , R 2, 61 2 4 18. F(6.5, 3.2), G(-5.1, 9.3)
(
) (
)
17. A -21, 0 , B -83, -61 2
4
4
19. X(-0.4, -4.8), Y(1.8, -8.8)
20. GEOMETRY Is ABC with vertices A(8, 4), B(-2, 7), and C(0, 9) a scalene triangle? Explain.
EXTRA
PRACTICE
21. DARTS Darnell’s first dart lands 2 inches to the right and 7 inches below the bull’s-eye. What is the distance between the bull’s-eye and where his first shot hit the target? Round to the nearest tenth of an inch.
See pages 782, 802. Self-Check Quiz at pre-alg.com
H.O.T. Problems
22. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the distance between two points. 23. OPEN ENDED Give the coordinates of the endpoints of a line segment that is neither horizontal nor vertical and has a length of 5 units. 24. CHALLENGE Find the values of x if the distance between (1, 2) and (x, 7) is 13 units. 25. SELECT A TECHNIQUE In a golf tournament, Joan’s ball landed 2 feet to the left and 3 feet short of the cup. Carolina’s ball landed 1 foot to the right and 4 feet beyond the cup. Which of the following techniques would you use to determine who is closer to the cup? Justify your selection(s). Then use the technique(s) to solve the problem. mental math
number sense
estimation
Lesson 9-5 The Distance Formula Dan Loh/AP/Wide World Photos
495
26.
Writing in Math How is the Distance Formula related to the Pythagorean Theorem? Include a comparison of the expressions (x2 - x1) and (y2 - y1) to the lengths of the legs of a right triangle.
27. Which expression shows how to find the distance between points M and N?
28. What is the distance between S and T in quadrilateral RSTU? Round to the nearest tenth.
Y
Y
4
. Ó] ή
3 X
"
- x] ή
X
"
2
A B C D
(2 - 5)2 + (3 - 3)2 √ [2 - (-5)]2 + [3 - (-3)]2 √ [2 - (-5)]2 + (3 - 3)2 √ (3 - 2)2 + [3 - (-5)]2 √
5
F 4.5
H 5.7
G 5.4
J 10.8
Find the length of the hypotenuse in each right triangle. Round to the nearest tenth, if necessary. (Lesson 9-4) 29.
30. 6 ft
31.
24 yd
c ft
12 km
9 yd c yd
c km
33 km
7 ft
32. ALGEBRA The measures of the angles of a triangle are in the ratio 1:4:5. Find the measure of each angle. (Lesson 9-3) 33. What number is 56% of 85? (Lesson 6-8) 34. SPACE The table shows the mass of 4 planets. About how many times bigger is Jupiter than Mercury? (Lesson 4-7)
0LANET
-ASS KG
%ARTH
*UPITER
-ERCURY
.EPTUNE -ÕÀVi\ .!3!
PREREQUISITE SKILL Solve each proportion. (Lesson 6-3) 84 _ a 4 7 12 35. _ =_ 36. _ =_ 37. _ =m x 16
15
60
496 Chapter 9 Real Numbers and Right Triangles
52
13
2.8 4.2 38. _ =_ h
12
9-6
Similar Figures and Indirect Measurement
Main Ideas • Identify corresponding parts and find missing measures of similar figures.
Have you ever used a copy machine to make an enlargement or reduction of a drawing? In this activity, you will draw enlargements and reductions. In mathematics, these are known as dilations.
• Solve problems involving indirect measurement using similar triangles.
Step 1 On grid paper, draw a rectangle whose length is 5 inches and whose width is 2 inches. This is the original figure.
New Vocabulary similar figures indirect measurement
IN IN
Step 2 Use a scale factor of 1.5. Draw a new rectangle whose length is 1.5 × 5 inches and whose width is 1.5 × 2 inches. a. What are the dimensions of the new rectangle? b. Is the new rectangle an enlargement or a reduction? c. Repeat Steps 1 and 2 with a right triangle whose legs measure 3 inches and 4 inches. 1 . Use a scale factor of _
IN
3
d. What are the dimensions of the new triangle?
IN
e. Is the new triangle an enlargement or a reduction? f. Compare the measures of the angles of each pair of figures. Do you notice any patterns? g. How do the lengths of the sides of the figures compare? Do you notice any patterns? h. MAKE A CONJECTURE Repeat Steps 1 and 2 using different figures and different scale factors. What kinds of scale factors result in an enlargement? A reduction?
Corresponding Parts Figures that have the same shape but not Reading Math Similar The symbol ∼ is read is similar to.
necessarily the same size are called similar figures. Figure ABCD is similar to figure EFGH. This is written as figure ABCD ∼ figure EFGH, with both sets of vertices listed in the same order. & " '
% !
# $
(
Similar figures have corresponding angles and corresponding sides. Arcs are used to show congruent angles. Lesson 9-6 Similar Figures and Indirect Measurement
497
Corresponding Parts of Similar Figures If two figures are similar, then • the corresponding angles have the same measure, and • the corresponding sides are proportional.
Words
Y
B
ABC ∼ XYZ
Model A
C
X
Z
BC AC AB ∠A ∠X, ∠B ∠Y, ∠C ∠ Z and _ = _ = _
Symbols
YZ
XY
XZ
Since corresponding sides are proportional, you can use a proportion or the direct variation equation to determine the measures of the sides of similar figures when some measures are known.
EXAMPLE BrainPOP® pre-alg.com
Find Measures of Similar Figures
The figures are similar. Find each missing measure. a.
J K
5 cm
b.
R
3 cm
2 Ón vÌ
M 9 cm
5
4
8 S
Scale Factor ratio of a length on a polygon to the corresponding length on a similar polygon (Lesson 6-4)
JM
x cm
{ vÌ
x = 15
Find the cross products.
y = kx
Simplify.
d = 1(49) Substitute. d=7
Mentally divide each side by 3.
1B.
)
, -
{ V (
x V
/ .
+ £{ V
Îä V
3 Ó{ V
,
498 Chapter 9 Real Numbers and Right Triangles
Multiply.
L V
Ç V
> V
Direct variation equation
7
£ä V
*
:
Replace JM with 3, RT with 9, KM with 5, and ST with x.
1A. '
9
Write a proportion.
3= 5 9 x
3x = 45
7
The scale factor that relates RSTU 4 or 1. Use the scale to MNOP is 7 28 factor to relate dimensions in MNOP, x, to dimensions in RSTU, y.
KM
= RT ST
3·x=9·5
` vÌ
T
The corresponding sides are proportional.
Review Vocabulary
3
{ vÌ
1
2 Extra Examples at pre-alg.com
Indirect Measurement The properties of similar triangles can be used to find measurements that are difficult to measure directly. This kind of measurement is called indirect measurement.
MAPS In the figure, ABE ∼ DCE. Find the distance across the lake.
Fox Lane
64 = 1(d) 2 Real-World Career Cartographer A cartographer gathers geographic, political, and cultural information and then uses this information to create graphic or digital maps of areas.
2(64) = d
Gazelle Road
64 yd
The scale factor that relates DCE to ABE or 1. is 48 96 2 y = kx
A E B 48 yd
C 96 yd
Direct variation equation.
d yd
Substitute. Multiply each side by 2.
D
128 = d XM
1
2. In the figure, STU ∼ VQU. Find the distance across the river.
6
M
5 3
For more information, go to pre-alg.com.
M
M 4
Personal Tutor at pre-alg.com
MEMORIALS The lead statue of the Korean War Memorial in Washington, D.C., casts a 43.5-inch shadow at the same time a nearby tourist casts a 32-inch shadow. If the tourist is 64 inches tall, how tall is the lead statue? Explore You know the lengths of the shadows and the height of the tourist. You need to find the statue’s height. Shadow Reckoning
Plan
Write and solve a proportion.
Solve
tourist’s height 64 32 = statue’s height 43.5 h
Using similar triangles to solve problems involving shadows is called shadow reckoning.
64 · 43.5 = 32 · h 2784 = 32h 87 = h
tourist’s shadow statue’s shadow Find the cross products. Multiply. Divide each side by 32.
The height of the statue is 87 inches or 7 feet 3 inches.
Check
The tourist’s height is 2 times the length of his or her shadow. The statue should be 2 times its shadow, or 2 · 43.5, which is 87 inches.
3. MONUMENTS Suppose a bell tower casts a 27.6-foot shadow at the same time a nearby tourist casts a 1.2-foot shadow. If the tourist is 6 feet tall, how tall is the tower? Lesson 9-6 Similar Figures and Indirect Measurement Geoff Butler
499
Example 1 (p. 498)
The figures are similar. Find each missing measure. 1.
2. *
R C
+
£Ó °
Ç °
6 ft
S
° ,
1
ÓÇ °
T
x ft
Ý °
9 ft
D
Example 2 (p. 499)
.
E
15 ft
3. MAPS In the figure, ABC ∼ EDC. Find the distance from Austintown to North Jackson.
North Jackson
C 6 km Ellsworth
A
(p. 499)
HOMEWORK
HELP
For See Exercises Examples 5–10 1 11, 12 3 13, 14 2
B
18 km
4. SHADOWS At the same time a 10-foot flagpole casts an 8-foot shadow, a nearby tree casts a 40-foot shadow. How tall is the tree?
The figures are similar. Find each missing measure. 5.
6.
Y
H
$ Óx
5 in. 10 in.
X
4 in.
Z
£ä
Ý
#
!
"
+ *
G
x in.
7. *
£Ó vÌ
Ý vÌ
/
,
&
8.
£x vÌ
'
n vÌ
J
M
È vÌ
.
Q K 6 in.
R 9 in.
9.
{
x in.
9 in.
(
£ä
C
R 2 km S K 7.5 km
D
3 ft
P
10.
4.2 ft
x km
E x ft
A 7 ft
B
U
9 km
V
For Exercises 11 and 12, write a proportion. Then determine the missing measure. 11. RIDES Suppose a roller coaster casts a shadow of 31.5 feet. At the same time, a nearby Ferris wheel casts a 19-foot shadow. A sign says the roller coaster is 126 feet tall. How tall is the Ferris wheel?
500 Chapter 9 Real Numbers and Right Triangles
, È
J
Óä vÌ
Similar triangles and indirect measurement are used to solve problems about the height of structures. Visit pre-alg.com to continue work on your project.
Austintown x km E
D
2 km Deerfield
Example 3
/
ÎÈ °
(
12. ANIMALS At the same time a baby giraffe casts a 3.2-foot shadow, a 15-foot adult giraffe casts an 8-foot shadow. How tall is the baby giraffe? 13. PARKS How far is the pavilion from the log cabin?
14. ZOO How far are the gorillas from the cheetahs? 84 m
lake fish 51 yd
gorillas
48 yd
xm cabin cheetahs
flower garden 64 yd
35 m reptiles
x yd
birds
28 m pavilion
Determine whether each statement is sometimes, always, or never true. Explain. 15. The measures of corresponding angles in similar figures are the same. 16. Similar figures have the same shape and the same size. Real-World Link The giraffe is the tallest of all living animals. At birth, the height of a giraffe is about 6 feet tall. An adult giraffe is about 18–19 feet tall. Source:
infoplease.com
17. GEOMETRY Triangle LMN is similar to RST. What is the value of LN if RT is 9 inches, MN is 21 inches, and ST is 7 inches? 2 , draw 18. Using a scale factor of _
4 , draw 19. Using a scale factor of _
and label the new image of rectangle ABCD.
and label the new image of triangle XYZ.
3
!
3
"
8
CM
$
IN
#
CM
9
IN
:
PERIMETER AND AREA For Exercises 20–23, the figures are similar. 20. Find the perimeter of both figures. 21. Compare the scale factor of the side lengths and the scale factor of the perimeters. Explain. 22. Find the area of both figures. 23. Compare the scale factor of the side lengths and the scale factor of the areas. Explain. 3
2 }ÕÀi £
5
4
vÌ
8
7 { vÌ
}ÕÀi Ó
:
È vÌ
9
24. SCALE FACTORS Figure FGHJK ∼ figure LMNPQ. The scale factor from figure 3 . What is the perimeter of figure LMNPQ? FGHJK to figure LMNPQ is _ EXTRA
PRACTICE
See pages 783, 802. Self-Check Quiz at pre-alg.com
& Ó{
2
{ä
,
+
' ÓÎ
Îä
*
ÎÈ
(
1
.
Lesson 9-6 Similar Figures and Indirect Measurement Adam Jones/Photo Researchers
501
25. OPEN ENDED Draw two similar triangles whose scale factor is 1. Justify 3 your answer.
H.O.T. Problems
AB where 26. FIND THE ERROR Carla and Tony are finding the length of ABC ∼ DEF, BC = 16 feet, EF = 12 feet, and DE = 18 feet. Who is correct? Explain your reasoning. Carla
Tony
16 x = 12 18
16 = 12 x 18
x = 24 ft
x = 13.5 ft
27. CHALLENGE Triangle ABC has side lengths of 3 inches, 5 inches, and 6 inches. Triangle DEF has side lengths 4 inches, 6 inches, and 8 inches. Determine whether ABC ∼ DEF. Explain. 28.
Writing in Math Suppose you have two triangles. Triangle A is similar to triangle B, and the measures of the sides of triangle A are less than the measures of the sides of triangle B. The scale factor is 0.25. Which is the original triangle? Explain.
29. Quadrilateral ABCD is similar to quadrilateral EFGH. What is the length ? of FG $
&
!
£Ó
%
n
# È
' (
"
A 1.5 m
C 4m
B 3.8 m
D 5.3 m
30. Two coordinates for RST are shown. Which coordinates for point T will make MNP and RST similar triangles? Y F T(3, 1)
-
G T(1, -1) H T(1, 2) J T(1, 1)
.
" 0
X
Find the distance between each pair of points. Round to the nearest tenth, if necessary. (Lesson 9-5) 31. S(2, 3), T(0, 6)
32. E(-1, 1), F(3, -2)
33. W(4, -6), V(-3, -5)
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. (Lesson 9-4) 34. a = 10, b = 15, c = ?
35. a = 8, b = ?, c = 34
36. a = ?, b = 27, c = 82
37. GEOMETRY If each side has a length of 1 foot, find the perimeter of a figure with 9 pentagons. (Lesson 3-8)
502 Chapter 9 Real Numbers and Right Triangles
2
3
CH
APTER
Study Guide and Review
9
wnload Vocabulary view from pre-alg.com
Key Vocabulary Be sure the following Key Concepts are noted in your Foldable.
3IGHT S 5RIANGLE
Key Concepts Squares and Square Roots
(Lesson 9-1)
• A square root of a number is one of two equal factors of the number.
The Real Number System
(Lesson 9-2)
• Numbers that cannot be written as terminating or repeating decimals are called irrational numbers. • The set of rational and the set of irrational numbers together make up the set of real numbers.
acute angle (p. 477)
radical sign (p. 464)
acute triangle (p. 478)
real numbers (p. 469)
congruent (p. 478)
right angle (p. 477)
converse (p. 487)
right triangle (p. 478)
equilateral triangle (p. 478)
scalene triangle (p. 478)
hypotenuse (p. 485)
similar figures (p. 497)
irrational numbers (p. 469)
solving a right triangle
isosceles triangle (p. 478)
(p. 486)
legs (p. 485)
square root (p. 464)
line segment (p. 476)
straight angle (p. 477)
obtuse angle (p. 477)
triangle (p. 476)
obtuse triangle (p. 478)
vertex (p. 476)
perfect square (p. 464)
Triangles (Lesson 9-3) • An acute angle has a measure between 0° and 90°.
Vocabulary Check Complete each sentence with the correct term. Choose from the list above.
• A right angle measures 90°. • An obtuse angle has a measure between 90° and 180°.
1. A(n) ___?_____ triangle has one angle with a measurement greater than 90°.
• A straight angle measures 180°.
2. A(n) ___?_____ has all sides congruent.
• Triangles can be classified by their angles as acute, obtuse, or right and by their sides as scalene, isosceles, or equilateral.
The Pythagorean Theorem •
c2
=
a2
+
(Lesson 9-4)
b2
The Distance Formula
3. In a right triangle, the side opposite the right angle is the ___?_____. 4. Figures that have the same shape but not necessarily the same size are called ___?_____. 5. A(n) ___?___ is a square of a whole number.
(Lesson 9-5)
2 2 • d = 兹(x 2 - x1) + (y2 - y1)
Similar Figures and Indirect Measurement (Lesson 9-6) • If two figures are similar, then the corresponding angles have the same measure, and the corresponding sides are proportional.
Vocabulary Review at pre-alg.com
6. A triangle with no sides congruent is ___?_____. 7. ___?_____ polygons have the same shape and the same size. 8. Decimals that do not repeat or terminate are called ___?_____. 9. The ___?_____ of a right triangle are adjacent to the right angle.
Chapter 9 Study Guide and Review
503
CH
A PT ER
9
Study Guide and Review
Lesson-by-Lesson Review 9–1
Squares and Square Roots
(p. 464–468)
Find each square root. 10. √ 36 11. √ 100 12. - √ 81 13. ± √ 121 14. √ 484 15. - √ 225 16. GARDENING Each tomato plant needs 3 square feet of space to grow. The gardener wants to have a garden that is a square and that is large enough for 27 tomato plants. What is the area of the garden?
49 . Example 1 Find - √ - √ 49 indicates the negative square root of 49. 49 , = -7. Since 72 = 49, - √ . Example 2 Find ± √256 ± √ 256 indicates both square roots of 256. 256 = 16 and Since 162 = 256, √ 256 = -16. - √
17. CLOCKS The period of a pendulum is the time required for it to make one complete swing back and forth. The formula of the period P of a pendulum , where is the length of is P = 2 32 the pendulum in feet. If a pendulum in a clock tower is 8 feet long, find the period. Use 3.14 for .
√
9–2
The Real Number System
(p. 469–474)
Solve each equation. Round to the nearest tenth, if necessary. 18. n2 = 81 19. t2 = 38 20. 4y2 = 5.76 21. 37.5 = 5r 2 22. SPORTS To find the height h in meters of an object hit into the air, use the formula h = -4.9t 2 + 30t + 1.4, where t is the time in seconds. What height would a baseball reach 3 seconds after it is hit? 23. BUSINESS Kyle owns a business selling baseball cards on the internet. The function y = x2 + 50x + 1800 models the profit y that Kyle has made in month x for the first two years of his business. What is Kyle’s profit in month 15? 504 Chapter 9 Real Numbers and Right Triangles
Example 3 Solve x 2 = 72. Round to the nearest tenth. x2 = 72 √x2 = √ 72
Write the equation. Take the square root of each side.
x = √ 72 or x = - √ 72
Find the positive and negative square root.
x ≈ 8.5 or x ≈ -8.5
Simplify.
Mixed Problem Solving
For mixed problem-solving practice, see page 802.
9–3
Triangles
(p. 476–481)
Example 4 Classify the triangle by its angles and by its sides.
Classify each triangle by its angles and by its sides. 25. 24. 132˚ 60˚ 24˚ 60˚
H 80˚ 24˚
60˚
J
80˚
K
26.
27. 56˚
Óx Èä
34˚
28. SIGNS Classify the yield sign by its angles and by its sides.
9–4
The Pythagorean Theorem
x
Triangle HJK has all acute angles and two congruent sides. So, HJK is an acute isosceles triangle.
YIELD
(p. 485–490)
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary.
Example 5 Find the missing measure of the right triangle. 22
29. a = 6, b = ?, c = 15
9
30. a = ?, b = 2, c = 7 31. a = 18, b = ?, c = 24 32. BASEBALL On a baseball diamond, the bases are 90 feet apart. What is the distance from home plate to second base in a straight line? 33. TRAVEL Tiananmen Square in Beijing, China is the largest town square in the world, covering 95 acres. One square mile is 640 acres. Assuming that Tiananmen Square is a square, how many feet long is a side to the nearest foot?
b
c2 = a2 + b2
Pythagorean Theorem
222 = 92 + b2
Replace c with 22 and a with 9.
484 = 81 + b2
Simplify.
403 = b2
Subtract 81 from each side.
20.1 ≈ b
Take the square root of each side.
Chapter 9 Study Guide and Review
505
CH
A PT ER
9 9–5
Study Guide and Review
The Distance Formula
(p. 492–496)
Example 6 Find the distance between A(-4, 0) and B(2, 5).
Find the distance between each pair of points. Round to the nearest tenth, if necessary.
(x2 - x1)2 + (y2 - y1)2 Distance Formula d = 兹
34. J(0, 9), K(2, 7) 35. A(-5, 1), B(3, 6) 36. W(8, -4), Y(3, 3) 37. G(0, 0), H(3, 4) 38. AIRPORTS A distance of 3 units on the grid equals an actual distance of 1 mile. Suppose the locations of two airports on a map are at (121, 145) and (218, 401). Find the actual distance between these airports to the nearest mile.
9–6
Similar Figures and Indirect Measurement In Exercises 39 and 40, the figures are similar. Find each missing measure. 39.
= 兹 [2 - (-4)]2 + (5 - 0)2
(x1, y1) = (-4, 0), (x2, y2) = (2, 5)
= 兹 (6)2 + 5 2
Subtract.
= 兹 61
Simplify.
≈ 7.8
Simplify.
(p. 497–502)
Example 7 If ABC ∼ KLM, what is the value of x?
M
24 m
L
21 m 8m
L
N
40.
B
U
V
A x in. C
B x ft
BC AC _ =_
6 ft
A
LM KM x _ = _2 3 4
C J 15 ft
H
2 in.
xm
T
4 in.
x = 1.5
9 ft
K
41. WORLD RECORDS At 7 feet 8 inches, the world’s tallest woman casts a 46-inch shadow. At the same time, the world’s shortest woman casts an 18-inch shadow. How tall is the world’s shortest woman? 506 Chapter 9 Real Numbers and Right Triangles
K
3 in.
M
Write a proportion. Substitution Find cross products and simplify.
CH
A PT ER
9
Practice Test 15. HIKING Brandon hikes 7 miles south and 4 miles west. How far is he from the starting point of his hike? Round to the nearest tenth.
Find each square root, if possible. 2. - √ 121 4. √ a2
1. √256 3. ± √ 49
5. Without using a calculator, estimate - √ 42 to the nearest whole number.
16. LADDER There is a building with a 12-foot high window. You want to use a ladder to reach the window. If the bottom of the ladder is 5 feet away from the building, will a 15-foot ladder reach the window? Explain.
ALGEBRA Solve each equation. Round to the nearest tenth, if necessary. 6. x2 = 100 7. w2 = 39
FT
8. 4.5u2 = 306
FT
For Exercises 9 and 10, use the triangle shown below. Find the distance between each pair of points. Round to the nearest tenth, if necessary.
-
17. A(3, 8), B(-5, 2) 18. Q(-6, 4), R(6, -8) .
nÇc
ÎÓc
19. C(5, 9), D(-7, 3)
9. Find the measure of ∠M. 10. Classify MNP by its angles and by its sides.
20. MULTIPLE CHOICE In the map of the park, the triangles are similar. Find the distance to the nearest tenth from the playground to the swimming pool.
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary.
45 ft
50 ft 36 ft
11. a = 6 yd, b = 8 yd, c = ?
Playground
12. a = 15 cm, b = ?, c = 32 cm 13.
Swimming Pool
x ft
14. V vÌ
£ä vÌ
V
Îä
A 60.5 ft B 62.5 ft
£Ó vÌ
C 63.1 ft £È
Chapter Test at pre-alg.com
D 64.2 ft
Chapter 9 Practice Test
507
CH
A PT ER
Standardized Test Practice
9
Cumulative, Chapters 1–9
Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. GRIDDABLE Triangle ABC is similar to N ? LMN. What is the length, in yards, of L
9 yd
N
C
2. Mrs. Hamilton wants to have a circular swimming pool installed in her backyard, but it would intersect part of an elevated pathway. Find x, the inside length of the section of pathway that would intersect the pool. x 20 ft
1
2
3
4
F point P
J point S
B
A
H point R
12 yd
L
PQ R S
G point Q
M
8 yd
3. Which point on the number line shown is 7? closest to √
16 ft
4. A salesperson earned a commission of $36 on a sale worth $800. Which statement below identifies a commission that is the same rate? A a commission of $60 on a $1200 sale B a commission of $77 on a $1400 sale C a commission of $27 on a $600 sale D a commission of $35 on a $750 sale 5. Molly multiplied her age by 2, subtracted 3, divided by 9, and added 4. The result was 7. Which could be the first step in finding Molly’s age? F Add 2 and 3. G Multiply 3 by 9. H Subtract 4 from 7. J Divide 7 by 4.
A 12 ft
C 28 ft
B 24 ft
D 32 ft
6. In the equation 2s - t = s + 3t, which would NOT be an appropriate first step to solve the 1? equation for t if s = 2 A Divide each side by 2s. 1. B Replace s with 2
C Add t to each side. Pace yourself Do not spend too much time on any one question. If you’re having difficulty answering a question, mark it in your test booklet and go on to the next question. Make sure that you also skip the question on your answer sheet. At the end of the test, go back and answer the question that you skipped.
508 Chapter 9 Real Numbers and Right Triangles
D Subtract s from each side. 7. GRIDDABLE Admission to the carnival is $6 and each ride is $0.50. If Hector wants to spend no more than $12 at the carnival, what is the maximum number of rides on which he can ride? Standardized Test Practice at pre-alg.com
Preparing for Standardized Tests For test-taking strategies and more practice, see pages 809–826.
4 7? 8. Which fraction is between and 3 F 4 5 G 6
12. GRIDDABLE Heather spent $172 on concert tickets. She bought 3 floor seats for $35.50 and 2 mezzanine seats. How much in dollars did the mezzanine seats cost?
8
5 8 H 9 11 J 12
Pre-AP
9. Stan has 40 baseball cards in his collection. He plans to add another 2 cards each week until he has doubled the amount in his collection. Which equation can be used to determine w, the number of weeks it will take to double the size of the baseball card collection? A B C D
13. The walls of a house usually meet to form a right angle. You can use string to determine whether two walls meet at a right angle. a. Copy the diagram shown below. Then illustrate the following situation.
40 × 2w = 80 2w + 40 = 80 w + 40 = 80 2w + 40 = 42
10. The table represents a function between x and y. What is the missing number in the table? F G H J
Record your answers on a sheet of paper. Show your work.
4 5 6 7
x
y
1
3
2
?
4
9
6
13
From a corner of the house, a 6-foot-long piece of string is extended along one side of the wall, parallel to the floor. From the same corner, an 8-foot-long piece of string is extended along the other wall, parallel to the floor. b. If the walls of the house meet at a right angle, then what is the distance between the ends of the two pieces of string? c. Draw an example of a situation where two walls of a house meet at an angle whose measure is greater than the measure of a right angle. d. Suppose the length of the walls in part c are the same length as the walls in part a, and that the 6-foot and 8-foot pieces of string are extended from the same corner. Do you think the distance between the two ends will be the same as in part b? Explain your reasoning.
11. One side of a garden is against a house as shown. If 88 feet of fencing will be used to enclose the garden, what is the width, w, of the garden? (2w 6) ft w
A 40 B 34
C 30 D 20
NEED EXTRA HELP? If You Missed Question...
1
2
3
4
5
6
7
8
9
10
11
12
13
Go to Lesson...
9-6
9-4
9-1
6-1
1-1
8-1
8-6
5-2
7-1
7-2
8-2
3-5
9-4
Chapter 9 Standardized Test Practice
509
10 •
Identify the relationships of parallel and intersecting lines.
•
Identify properties of congruent triangles.
• •
Identify and draw transformations.
•
Find the area of polygons and irregular figures, and find the area and circumference of circles.
Two-Dimensional Figures
Classify and find angle measures of polygons.
Key Vocabulary circle (p. 551) composite figures (p. 558) parallel lines (p. 512) transformation (p. 524)
Real-World Link Art The noses of 10 graffiti-covered Cadillacs are halfburied in a field west of Amarillo, Texas. The angles that the cars make with the ground are corresponding angles.
Compare and Contrast Polygons Make this Foldable to help you organize information about the characteristics of two-dimensional figures. Begin with four plain sheets of 11˝ × 17˝ paper, eight index cards, and glue.
1 Fold in half widthwise.
2 Fold the bottom to form a pocket. Glue the edges.
3 Repeat three times. Then glue all four pieces together to form a booklet.
510 Chapter 10 Two-Dimensional Figures Robyn Beck/Getty Images
4 Label each pocket. Place an index card in each pocket. LE S 4RAPE 4R IA N G ZOIDS E
GET READY for Chapter 10 Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2 Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Solve each equation. (Lesson 3-3) 1. x + 46 = 90 2. x + 35 = 180
Example 1
Solve 7x - 2 = -72.
3. 2x - 12 = 90
4. 3x - 24 = 180
7x - 2 = -72
Write the equation.
5. 5x + 165 = 360
6. 4x + 184 = 360
7x - 2 + 2 = -72 + 2
Add 2 to each side.
7. TRAVEL Ernesto drove the same number of miles each day from Monday through Friday, and 26 miles on the weekend. If he drove a total of 176 miles during the week, how many miles does he drive each workday? (Lesson 3-3)
Find each product. Round to the nearest tenth, if necessary. (Prerequisite Skills, pp. 747–748) 8. (5.5)(8) 9. (7.5)(3.4) 10. (6.3)(11.4)
1 11. _ (8)(2.5)
12. (2)(3.14)(1.7)
13. 2(3.1)(3.14)
2
7x = - 70
Simplify.
7x _ _ = -70
Divide each side by 7.
7
7
x = -10
Simplify.
Example 2
Find (0.5)(3)(6.25). Round to the nearest tenth. (0.5)(3)(6.25) = [(0.5)(3)](6.25) Multiply 0.5 × 3 first.
14. FOOD Nicole poured 32.5 bowls of soup volunteering her time in a soup kitchen. If each bowl contained 16.5 ounces of soup, how much soup did Nicole pour?
= 9.375
Simplify.
≈ 9.4
Round to the nearest tenth.
(Prerequisite Skills, pp. 747–748)
Find each sum. (Lesson 5-7) 3 1 2 1 15. 5_ + 4_ 16. 2_ + 3_ 2 3 3 1 17. 1_ + 2_ 8 2 2 _ _ 19. 2 + 3 5 3 9
3 4 5 1 18. 6_ + 1_ 6 4 2 _ _ 20. 5 + 3 4 3 5
5 pounds 21. RECYCLING The class collected 12_ 6 1 _ of bottles and 8 pounds of aluminum 8
cans. How many pounds of glass and aluminum cans did the class collect? (Lesson 5-7)
Example 3 3 5 Find 1_ + 4_ .
6 4 3 5 29 7 1 _ + 4_ = _ +_ 6 6 4 4 3 29 2 7 _ =_ · +_·_ 6 2 4 3 58 21 =_ +_ 12 12 79 =_ 12 7 = 6_ 12
Write as improper fractions. Rename using the LCD, 12. Simplify. Add the numerators. Simplify.
Chapter 10 Get Ready for Chapter 10
511
10-1
Line and Angle Relationships
Main Ideas • Identify the relationships of angles formed by two parallel lines and a transversal.
A satellite dish receives signals from a satellite and directs them into a receiver. The intersecting lines from the signal lines form different angle relationships.
• Identify the relationships of vertical, adjacent, complementary, and supplementary angles.
a. What do you notice about the lines coming into the satellite dish?
New Vocabulary parallel lines transversal interior angles exterior angles alternate interior angles alternate exterior angles corresponding angles vertical angles adjacent angles complementary angles supplementary angles perpendicular lines
2
1 3 4
5 6 8 7
b. Trace the red lines onto a piece of paper. Find the measure of each numbered angle. c. What do you notice about the measures of the angles? Which angles have the same measure?
Parallel Lines and a Transversal In geometry, two lines in a plane that never intersect are parallel lines. Lines m and n are parallel. Using symbols, m n.
m
Parallel lines have no point of intersection.
n
When two parallel lines are intersected by a third line called a transversal, eight angles are formed.
Names of Special Angles The eight angles formed by parallel lines and a transversal have special names. • Interior angles lie inside the parallel lines. ∠3, ∠4, ∠5, ∠6 • Exterior angles lie outside the parallel lines. ∠1, ∠2, ∠7, ∠8 • Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. ∠3 and ∠5, ∠4 and ∠6
READING in the Content Area For strategies in reading this lesson, visit pre-alg.com.
1
2 4
3 5
6 8
7
Arrowheads are often used in figures to indicate parallel lines.
• Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. ∠1 and ∠7, ∠2 and ∠8 • Corresponding angles are in the same position on the parallel lines in relation to the transversal. ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
512 Chapter 10 Two-Dimensional Figures Peter Pearson/Stone/Getty Images
transversal
In Lesson 9-3, you learned that line segments are congruent if they have the same measure. Similarly, angles are congruent if they have the same measure. Parallel Lines Cut by a Transversal If two parallel lines are cut by a transversal, then • corresponding angles are congruent, • alternate interior angles are congruent, and • alternate exterior angles are congruent.
EXAMPLE
Find Measures of Angles
In the figure at the right, m n and s and t are transversals. If m∠1 = 68°, find m∠5 and m∠6.
m
Since ∠1 and ∠5 are corresponding angles, they are congruent. So, m∠5 = 68°.
n
1 3 7 4
Since ∠1 and ∠6 are alternate exterior angles, they are congruent. So, m∠6 = 68°.
9 11 15 12
5 2 8 6 13 10
s t
16 14
1. If m∠11 = 84°, find m∠10 and m∠16.
Reading Math Congruent Angles Angle 1 is congruent to angle 2. This is written ∠1 ∠2. The measure of ∠1 is equal to the measure of ∠2. This is written m∠1 = m∠2.
Intersecting Lines and Angles Other pairs of angles have special relationships. When two lines intersect, they form two pairs of opposite angles called vertical angles. Vertical angles are congruent. The symbol for is congruent to is . ∠1 and ∠2 are vertical angles. ∠1 ∠2
1
3 4
2
∠3 and ∠4 are vertical angles. ∠3 ∠4
When two angles have the same vertex, share a common side, and do not overlap, they are adjacent angles. m∠AOB = m∠1 + m∠2
∠1 and ∠2 are adjacent angles.
A 2
1
B
O
If the sum of the measures of two angles is 90°, the angles are complementary.
4 3 1 2
m∠1 = 50°, m∠2 = 40° m∠1 + m∠2 = 90° Extra Examples at pre-alg.com
m∠3 = 60°, m∠4 = 30° m∠3 + m∠4 = 90° Lesson 10-1 Line and Angle Relationships
513
If the sum of the measures of two angles is 180°, the angles are supplementary.
1
2
3
4
m∠1 = 140°, m∠2 = 40° m∠1 + m∠2 = 180°
m∠3 = 45°, m∠4 = 135° m∠3 + m∠4 = 180°
Lines that intersect to form a right angle are perpendicular lines.
right angle
TILING Jun cuts a piece of tile at a 135° angle. What is the measure of the other angle formed by the cut?
Xª
ª
The angles at the cut point are supplementary. m∠x + 135 = 180
Write the equation.
m∠x + 135 - 135 = 180 - 135 m∠x = 45°
Subtract 135 from each side. Simplify.
2. ARCHITECTURE In the semicircular window, ∠1 is complementary to ∠2. If m∠2 is 24°, find m∠1.
Personal Tutor at pre-alg.com
EXAMPLE
Find Measures of Angles
ALGEBRA Angles ABC and FGH are complementary. If m∠ABC = x + 8 and m∠FGH = x - 10, find the measure of each angle. Step 1 Find the value of x. Check your Answer To check your answer, add to see if the sum of the measures of the angles is 90. Since 54 + 36 = 90, the answer is correct.
m∠ABC + m∠FGH = 90 Complementary angles (x + 8) + (x - 10) = 90 2x - 2 = 90 2x = 92 x = 46
Substitution Combine like terms. Add 2 to each side. Divide each side by 2.
Step 2 Replace x with 46 to find the measure of each angle. m∠ABC = x + 8
m∠FGH = x - 10
= 46 + 8 or 54
= 46 - 10 or 36
So, m∠ABC = 54° and m∠FGH = 36°.
3. ALGEBRA Angles MNO and RST are supplementary. If m∠MNO = 5x and m∠RST = x - 6, find the measure of each angle. 514 Chapter 10 Two-Dimensional Figures
SAFETY A lifeguard chair is shown. If m∠1 = 105°, find m∠4 and m∠6. Since ∠1 and ∠4 are vertical angles, they are congruent. So, m∠4 = 105°.
1 6 7 4
Since ∠6 and ∠1 are supplementary, the sum of their measures is 180°.
3 8 5 2
180 - 105 = 75. So, m∠6 = 75°.
4. Find m∠2 and m∠3 in the lifeguard chair above. Explain your reasoning.
Line and Angle Relationships
Parallel Lines
Perpendicular Lines
Vertical Angles
n 1
ab
a
m ⊥n
m
2 4
∠1 ∠3
3
∠2 ∠4
b
Adjacent Angles
Reading Math
1
A
1
2
(pp. 513, 515)
1
2
B
m∠ABC = m∠1 + m∠2
Examples 1 and 4
Supplementary Angles
C
D
Adjacent Angles Angles do not have to be adjacent to be complementary or supplementary angles.
Complementary Angles
m∠1 + m∠2 = 90°
2
m∠ + m∠2 = 180°
In the figure at the right, m and k is a transversal. k If m∠1 = 56°, find the measure of each angle. 1. ∠2
2. ∠3
ᐉ
3. ∠4
1
2 3
Example 2 (p. 514)
4.
5. 140˚
(p. 514)
4
Find the value of x in each figure. x˚
Example 3
m
152˚
6. x˚
X ÓÈ
7. ALGEBRA If m∠N = 3x and m∠M = 2x and ∠M and ∠N are supplementary, what is the measure of each angle? Lesson 10-1 Line and Angle Relationships
515
HOMEWORK
HELP
For See Exercises Examples 8–13 1, 4 14–21 2 22, 23 3
g
In the figure at the right, g h and t is a transversal. If m∠4 = 53°, find the measure of each angle. 8. ∠1
9. ∠5
10. ∠7
11. ∠8
12. ∠2
13. ∠3
4
7
8
6
t
1
3
2
5
h
Find the value of x in each figure. 14.
15. 45˚
x˚
16.
148˚ x˚
31˚ x˚
17.
x˚
18. 5˚
19. 8x ˚
4x ˚
5x ˚ 5x ˚
20. Find m∠A if m∠B = 17° and ∠A and ∠B are complementary. 21. Angles P and Q are supplementary. Find m∠P if m∠Q = 139°. 22. ALGEBRA Angles J and K are complementary. If m∠J = x - 9 and m∠K = x + 5, what is the measure of each angle? 23. ALGEBRA Find m∠E if ∠E and ∠F are supplementary, m∠E = 2x + 15, and m∠F = 5x - 38. ALGEBRA In the figure at the right, m and t is a transversal. Find the value of x for each of the following.
t
ᐉ
1 4
m
24. m∠2 = 2x + 3 and m∠4 = 4x - 7
5 8
2
3
6
7
25. m∠8 = 4x - 32 and m∠5 = 5x + 50 26. CONSTRUCTION To measure the angle between a sloped cathedral ceiling and a wall, a carpenter uses a plumb line (a string with a weight attached) as shown. If m∠YXB = 68°, what is m∠XBC? Explain your reasoning. 27. ALGEBRA The measure of the supplement of an angle is 15° less than four times the measure of the complement. Find the measure of the angle.
A
X B
Y
C
7
TIME For Exercises 28 and 29, use the clock showing 6 o’clock and 10 seconds. 28. Find m∠WXY and m∠YXZ. 29. Find the time that will show m∠WXY + m∠YXZ = 90°. 516 Chapter 10 Two-Dimensional Figures
9 8 :
30. ALGEBRA Angles R and S are complementary. The ratio of their measures is 4:5. Find the measure of each angle. ANALYZE GRAPHS For Exercises 31–33, use the graphs. y
y y 3x 2 y 2x 3 O
EXTRA
3
y 1x 1 2
See pages 783, 803.
H.O.T. Problems
x
O y 1x 1
PRACTICE
Self-Check Quiz at pre-alg.com
x
31. How are each pair of graphs related? 32. What seems to be true about the slopes of the graphs? 33. MAKE A CONJECTURE about the slopes of the graphs of perpendicular lines. 34. OPEN ENDED Draw a pair of adjacent, supplementary angles. 35. CHALLENGE Suppose two parallel lines are cut by a transversal. How are the interior angles on the same side of the transversal related? 36.
Writing in Math
How are parallel lines and angles related? Illustrate with a drawing of parallel lines intersected by a transversal and a list of the congruent and supplementary angles.
37. The hedge shears have two different sets of angles. Find x.
38. Which angles are NOT supplementary?
A 32
#
B 58
$
" x°
C 102 D 121
! 59°
:
%
F ∠EZC, ∠CZA
H ∠AZB, ∠BZE
G ∠BZC, ∠CZD
J
∠DZE, ∠AZD
39. FARMING At the same time a 40-foot silo casts a 22-foot shadow, a fence casts a 3.3-foot shadow. Find the height of the fence. (Lesson 9-6) 40. ARCHAEOLOGY Two artifacts are found at a dig. If a coordinate plane is set up, one artifact was found at (1, 5) and the other artifact was found at (3, 1). How far apart were the two artifacts? Round to the nearest tenth. (Lesson 9-5) ALGEBRA Solve each inequality. (Lesson 8-5) a 42. _ >3
41. 5m < 5
43. -4x ≥ -16
-2
PREREQUISITE SKILL Use a protractor to draw an angle having each measurement. (pp. 757–758) 44. 20°
45. 45°
46. 65°
47. 145°
48. 170°
Lesson 10-1 Line and Angle Relationships
517
10-2 Main Idea • Identify congruent triangles and corresponding parts of congruent triangles.
Congruent Triangles
Ivy is a type of climbing plant. Most ivy leaves have five major veins. In the photo shown, the outlines form two triangles.
New Vocabulary congruent corresponding parts
a. Trace the triangles shown at the right onto a sheet of paper. Then label the triangles.
C
E
B
F
b. Measure and then compare the lengths of the sides of the triangles. c. Measure the angles of each triangle. How do the angles compare?
D
A
d. Make a conjecture about the triangles.
Congruent Triangles Figures that have the same size and shape are congruent. The parts of congruent triangles that “match” are corresponding parts.
Vocabulary Link Corresponding Everyday Use having the same relationship Math Use having the same position
Corresponding Parts of Congruent Triangles Words
If two triangles are congruent, their corresponding sides are congruent and their corresponding angles are congruent.
Model
1
9
Tick marks are used to indicate which sides are congruent.
Arcs are used to indicate which angles are congruent.
8
:
2
Symbols Congruent Angles: ∠X ∠P, ∠Y ∠Q, ∠Z ∠R −− −− −− −− −− −− Congruent Sides: XY PQ, YZ QR, XZ PR
518 Chapter 10 Two-Dimensional Figures SuperStock
When writing a congruence statement, the letters must be written so that corresponding vertices appear in the same order. For example, for the diagram below, write FGH JKM. G
K
FGH J K M
Congruence Statements You can also write a congruence statement
F
H
J
Vertex F corresponds to vertex J. Vertex G corresponds to vertex K. Vertex H corresponds to vertex M.
M
as GHF KMJ, HFG MJK, FHG JMK, GFH KJM, and HGF MKJ.
EXAMPLE
Name Corresponding Parts
Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.
A
Z
Corresponding Angles B
∠A ∠Z, ∠B ∠Y, ∠C ∠X
C
X
Y
ABC ?
Corresponding Sides −− −− −− −− −− −− AB ZY, BC YX, CA XZ One congruence statement is ABC ZYX. &
2
1. Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement DEF ? . $
EXAMPLE
%
1
Identify Congruent Triangles
Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement.
R 20 m
16 m
Explore The drawing shows which S angles are congruent and the lengths of all sides.
U
24 m
20 m
T
W
24 m
16 m
V
Plan
Note which segments have the same length and which angles are congruent. Write corresponding vertices in the same order.
Solve
Angles: The arcs indicate that ∠S ∠W, ∠R ∠V, and ∠T ∠U. −− −−− WV , RT Sides: The side measures indicate that SR VU, and −−− TS UW. Since all pairs of corresponding angles and sides are congruent, the two triangles are congruent. One congruence statement is SRT WVU. (continued on the next page)
Extra Examples at pre-alg.com
Lesson 10-2 Congruent Triangles
519
Check Draw SRT and WVU so that they are oriented in the same way. Then compare the angles and sides.
R 16 m
S
V 16 m
20 m
24 m
T
W
20 m
24 m
U
Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 2A. ) * IN + IN
IN IN
IN
( IN '
,
2B. !
$
Èä
Èä Îä
"
#
&
Îä
%
You can use corresponding parts to find the measures of angles and sides in a figure that is congruent to a figure with known measures.
LANDSCAPING A brace is used to support a tree and to help it grow straight. In the figure, TRS ERS. a. At what angle is the brace placed against the ground?
∠E and ∠T are corresponding angles. So, they are congruent. Since m∠T = 65°, m∠E = 65°.
R brace
The brace is placed at a 65° angle with the ground.
8 ft
b. What is the length of the brace? −− −− and RE corresponds to RT. So, RE RT are congruent. Since RT = 8 feet, RE = 8 feet.
65˚
T
E S 3 ft
The length of the brace is 8 feet.
3. QUILTING A quilt design is shown. In the figure, ABC ADE. What is the measure of ∠BCA? What is the perimeter of the design?
# Personal Tutor at pre-alg.com
520 Chapter 10 Two-Dimensional Figures
Î V
£n V
!
$
"
xc
%
Example 1 (p. 519)
For each pair of congruent triangles, name the corresponding parts. Then complete the congruence statement. 1.
K
G
2. A
C
C B
J
M
B E
D
CBE ?
KMJ ? Example 2 (pp. 519–520)
Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 3.
!
4. #
$
&
£Ó°x vÌ
.
£x vÌ
£n vÌ
* 1 "
Example 3 (p. 520)
&
#
7
£x vÌ
%
5. TOWERS A tower that supports highvoltage power lines is shown at the right. In the tower, ADC BFC. FC if AC = 10 feet What is the length of and DC = 15 feet?
A
B C
D
HOMEWORK
HELP
For See Exercises Examples 6–9 1 10–13 2 14, 15 3
F
For each pair of congruent triangles, name the corresponding parts. Then complete the congruence statement. 6.
D
E
F
DEF
7. K
P
?
Q
8. A
J
R
. D
N
M
P
KJM
?
9.
Z
. R
B E
C
T
W
S Y
DBA
?
.
ZWY
?
.
Lesson 10-2 Congruent Triangles
521
Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. 10.
11.
P
0 .
Z
M
S
W
-
M
M M
M
M 2
1
/
Q V
12.
R 4 cm S F
6 cm 21˚
6 cm
G 9 cm
4 cm 32˚ H
Real-World Link Trusses were used in the construction of the Eiffel Tower in Paris, France. The tower contains more than 15,000 pieces of steel and 2.5 million rivets. Source: paris.org
13. (
+
104˚
8 cm
29˚
Q
*
4
R ARCHITECTURE For Exercises 14 and center top left chord 15, use the diagram of the roof truss web metal plate at the right. In the figure, TRU connector SRU. 30˚ T 14. Find the distance from the right U metal plate connector to the center bottom chord web. 32 ft 15. What is the measure of the angle formed by the top left chord and the bottom chord?
S
Find the value of x for each pair of congruent triangles. 16. A
12
D
17.
C 16
x
3
2 *
X
20
B
E
4 +
-
Determine whether each statement is true or false. If false, give a counterexample. 18. If two triangles are congruent, then the perimeters are equal. 19. If the perimeters of two triangles are equal, then the triangles are congruent. 20. ALGEBRA If ABC XYZ, what is the value of x?
A
Y 3x ⫹ 12
18
Z B EXTRA
PRACTICE
See pages 783, 803. Self-Check Quiz at pre-alg.com
21. BUTTERFLIES Butterfly wings are triangular in shape. Using the photograph of the butterfly as a model, draw two sets of congruent triangles, label the vertices, and write a congruence statement for each.
522 Chapter 10 Two-Dimensional Figures (cl)John Lawrence/Imagestate; (br)Burke/Triolo Productions/Brand X Pictures/Getty Images
C
24
X
30
H.O.T. Problems
22. FIND THE ERROR Jade and Fernando are writing a congruence statement for the congruent triangles at the right. Who is correct? Justify your reasoning.
3
4 8
2 :
9
Fernando −− −− ST YZ
Jade ∠YXZ ∠STR
D
23. CHALLENGE In the figure at the right, there are two pairs of congruent triangles. Write a congruence statement for each pair. 24.
Explain where congruent triangles are present in nature. Include a definition of congruent triangles and an example of an object in nature that contains congruent triangles.
10
Y U
A
I
H
F
G
F 12 ft
2
G 24 ft H 48 ft
24
Z
A 10
K
26. Guy wires create two congruent triangles PQR and SQR. Find the −− length of QS.
W 26
E J
B
Writing in Math
−− 25. Find the measure of UY if XYZ UYW.
X
C
J 65 ft B 20
C 24
D 26
1
3
FT
27. Angles P and Q are supplementary. Find m∠P if m∠Q = 129°. (Lesson 10-1) −− 28. Find AC if ABC ∼ DEF, AB = 15, DE = 10, and DF = 5. (Lesson 9-6) 29. GEOMETRY The table shows how the perimeters of an equilateral triangle and a square change as side lengths increase. Compare the rates of change. (Lesson 8-5) 30. FOOD A recipe for butter cookies requires 12 tablespoons 12 as a of sugar for every 16 tablespoons of flour. Write _ 16 fraction in simplest form. (Lesson 4-4)
Perimeter y
Side Length x
Triangle
Square
2
6
8
4
12
16
PREREQUISITE SKILL Graph each point on a coordinate plane. (Lesson 2-6) 31. A(2, 4) 32. J(-1, 3) 33. H(0, 5) 34. D(2, 0) 35. W(-2, -4)
Lesson 10-2 Congruent Triangles
523
10-3
Transformations on the Coordinate Plane
Main Idea • Draw translations, reflections, and dilations on a coordinate plane.
The physical motions used in recreational activities such as skateboarding or riding a scooter are related to mathematics.
New Vocabulary transformation image translation reflection line of symmetry dilation center
a. Describe the motion involved in making a 180° turn on a skateboard. b. What type of motion does a scooter display when moving forward?
Transformations A mapping of a geometric figure that may change its shape or position is a transformation. Every corresponding point on the figure after a transformation is called its image. Three types of transformations are shown below. Translation • In a translation, you slide a figure from one position to another without turning it. Translations are also called slides.
y
x
O
Reflection
• In a reflection, you flip a figure over a line. This line is called a line of symmetry. The figures are mirror images of each other. Reflections are also called flips.
y
line e of symmetry y e
O
x
$ILATION
Look Back To review scale factors, see Lesson 6-4.
• In a dilation, you enlarge or reduce a figure by a scale factor with respect to a fixed point called the center. The resulting image is similar to the original figure. The scale factor in the graph at the right is 3.
Y
CENTER "
524 Chapter 10 Two-Dimensional Figures
X
Reading Math Notation The notation M’ is read M prime. It corresponds to point M.
When translating a figure, every point of the original figure is moved the same distance and in the same direction. The image is congruent to the original figure and the orientation is the same as the original figure. Translation 5 units down
Translation 4 units left y
y
A
y
R'
B
X
X'
Translation 6 units right, 3 units up
R Y'
D
Y
A'
x
O
x
O C
O
x
T'
B'
Z
Z'
S' T
D'
C'
S
To translate a point in the coordinate plane, describe the translation using an ordered pair. Then add the coordinates of the ordered pair to the coordinates of the original point.
y 5 4 3 N 2 1
Triangle MNP is shown on the coordinate plane. Find the coordinates of the vertices of the image of MNP translated 5 units left and 3 units up. A B C D
M'(-1, 1), N'(0, 5), P'(-5, 5) M'(4, 1), N'(0, 5), P'(5, 5) M'(-1, 1), N'(-5, 5), P'(0, 5) M'(-1, -2), N'(-5, 2), P'(0, 2)
⫺4⫺3⫺2⫺1O
P 1 2 3 4x
⫺2 ⫺3
M
Read the Test Item Common Misconception In a translation, the order in which a figure is moved does not matter. For example, moving 3 units down and then 2 units right is the same as moving 2 units right and then 3 units down.
This translation can be written as (-5, 3). To find the coordinates of the translated image, add -5 to each x-coordinate and add 3 to each y-coordinate. Solve the Test Item vertex M(4, -2) N(0, 2) P(5, 2)
5 left, 3 up
+ + +
(-5, 3) (-5, 3) (-5, 3)
translation
→ → →
M’(-1, 1) N’(-5, 5) P’(0, 5)
The answer is C.
1. Triangle ABC is translated so that B is mapped to B’. Which coordinate pair best represents C’? F (-4, 1)
H (-1, 1)
G (0, 3)
J (1, 3)
"g
Y È x " { Î Ó £
#
!
{ÎÓ£ " £ Ó Î { X £ Ó
Personal Tutor at pre-alg.com Lesson 10-3 Transformations on the Coordinate Plane
525
When reflecting a figure, every point of the original figure has a corresponding point on the other side of the line of symmetry. The image is congruent to the original figure, but the orientation is different from the original figure. To reflect a point over the x-axis, use the same x-coordinate and multiply the y-coordinate by -1. To reflect a point over the y-axis, use the same y-coordinate and multiply the x-coordinate by -1.
Reflection over the x-axis y
Reflection over the y-axis y
E
x
O
I'
E'
EXAMPLE
H
H' F F' x
G O G'
I
J' J
Reflection in a Coordinate Plane
The vertices of a figure are A(-2, 3), B(0, 5), C(3, 1), and D(3, 3). Graph the figure and the image of the figure after a reflection over the x-axis. To find the coordinates of the vertices of the image after a reflection over the x-axis, use the same x-coordinate and multiply the y-coordinate by -1.
B
opposite same
A(-2, 3) B(0, 5) C(3, 1) D(3, 3)
→ → → →
(-2, -1 · 3) (0, -1 · 5) (3, -1 · 1) (3, -1 · 3)
D
A C C' x
O
→ → → →
A'(-2, -3) B'(0, -5) C'(3, -1) D'(3, -3)
A' D'
B'
2. The vertices of polygon DEFG are D(4, -2), E(5, -5), F(2, -4), and G(1, -1). Graph the polygon and the image of the figure after a reflection over the y-axis. The diagrams below show how dilations result in similar figures that are larger and smaller than the original. The center of each dilation is the origin. Scale Factor ⴝ 2 (enlargement) Vocabulary Link Dilate Everyday Use to expand or widen Math Use to enlarge or reduce by a scale factor
3CALE FACTOR
y
Y
B'
REDUCTION
9 : 9g
A'
:g
B C'
A C O
D
D'
X
"
x
8g 8
Suppose k is the scale factor. • If k > 1, the dilation is an enlargement. • If 0 < k < 1, the dilation is a reduction. • If k = 1, the dilation is congruent to the original figure. When the center of a dilation is the origin, you can find the coordinates of the image by multiplying the coordinates of a polygon by the scale factor. 526 Chapter 10 Two-Dimensional Figures
Extra Examples at pre-alg.com
EXAMPLE
Dilation in a Coordinate Plane
A figure has vertices J(2, 4), K(2, 6), M(8, 6), and N(8, 2). Graph the figure and the image of the figure after a dilation centered at the 1 origin with a scale factor of _ .
Y
To dilate the polygon, multiply the coordinates
+g
1 . of each vertex by _
*g
+
2
* -g
2
Corresponding Parts Dilated figures have congruent angles and sides that are proportional.
-
J(2, 4) → J’(1, 2)
K(2, 6) → K’(1, 3)
M(8, 6) → M’(4, 3)
N(8, 2) → N’(4, 1)
. .g X
"
3. A figure has vertices R(-1, 2), S(1, 4), and T(1, 1). Graph the figure and the image of the figure after a dilation centered at the origin with a scale factor of 3. Transformations Translations and Reflections produce images that are the same shape and the same size. The figures are congruent to the images. Dilations produce images that are similar (same shape, but not the same size). The figures are not congruent to the images, except when the scale factor k = 1.
Example 1 (p. 525)
1. MULTIPLE CHOICE Rectangle RSTU has been translated. Which describes the translation?
Y
2
3
A 4 units left, 2 units up B 4 units right, 2 units down
3g
2g
X
4 "
5
C 2 units right, 4 units down
4g
5g
D 2 units left, 4 units up Example 2 (p. 526)
2. Suppose the figure graphed is reflected over the y-axis. Find the coordinates of the vertices after the reflection. 3. Hold your hands in front of you with your palms down. What kind of transformation exists from your left hand to your right hand?
Example 3 (p. 527)
4. Triangle ABC is shown. Graph the image of ABC after a dilation centered at the origin with a scale factor of 2.
X y W
x
O
Y Z
Y
" ! X
"
#
Lesson 10-3 Transformations on the Coordinate Plane
527
HOMEWORK
HELP
For See Exercises Examples 5–8 1 9–12 2 13–18 3
Find the vertices of each figure after the given translation. Then graph the translation image. 5. (2, 3)
1 7. 5, 2_
(
6. (-4, 3) y
2
) y
y
E
A D H
B x
O
x
O
x
O
J C
F
I
K
G
8. The vertices of a figure are D(1, 2), E(1, 4), F(-4, 4), and G(-1, 2). Graph the image of the figure after a translation 4 units down. Find the vertices of each figure after a reflection over the given axis. Then graph the reflection image. 9. x-axis
10. x-axis
11. y-axis
y
R
y
y
N
S
B
A M O C x
O
P x
OQ
x
O
D
T
12. The vertices of a figure are W(-3, -3), X(0, -4), Y(4, -2), and Z(2, -1). Graph the image of its reflection over the y-axis. Find the vertices of each figure after a dilation with the given scale factor centered at the origin. Then graph the dilation image. 13. scale factor: 4
14. scale factor: 1.5
1 15. scale factor: _
Y
Y
Y
£{ £Ó £ä n È { Ó
( 1 0
' )
2
"
X
"
*
X
For Exercises 16–18, use the graph at the right. 16. Graph the image of the figure after a dilation with a scale factor of 2 with the center at the origin. 17. Graph the image of the original figure after a dilation 1 with the center at the origin. with a scale factor of _
4
9 8 : Ó { È n £ä £Ó £{X
"
Y
8 :
"
X
2
18. Find the vertices of the original figure after a dilation with a scale factor of 1.5 with the center at the origin. 528 Chapter 10 Two-Dimensional Figures
9
19. BIOLOGY A microscope dilates the image of objects by a scale factor of 12. How large will a 0.0016-millimeter paramecium appear? 20. GAMES What type of transformation is used when moving a knight in a game of chess? Explain. 21. MIRRORS Which transformation exists when you look into a mirror? Explain.
Real-World Link In chess, each player has 16 game pieces, or chessmen. There are two rooks, two knights, two bishops, a queen, a king, and eight pawns.
22. Give a counterexample for the following statement. The image of a figure’s reflection is never the same as the image of its translation. 23. PRESENTATIONS Felicia wants to project a 2-inch by 2-inch slide onto a wall to create an image 128 inches by 128 inches. If the slide projector makes the image twice as large for each yard that it is moved away from the wall, how far away should Felicia place the projector? ANALYZE GRAPHS Identify each transformation as a translation, a reflection, or a dilation. 25. 26. 24. Y Y Y
EXTRA
PRACTICE
See pages 784, 803.
"
X
"
X
X
"
Self-Check Quiz at pre-alg.com
H.O.T. Problems
27. OPEN ENDED Draw a triangle on grid paper. Then draw the image of the 1 . triangle after it is moved 5 units right and then dilated by a scale factor of _ 3
Y
28. CHALLENGE Suppose figure ABCD is dilated by a scale factor of 2 and then reflected over the x-axis and the y-axis. Describe the resulting figure and explain how it relates to a reflection. Then graph the image A’B’C’D’ on a coordinate plane.
" #
!
$ X
"
29. Which One Doesn’t Belong? Without graphing, identify the pair of points that does not represent a reflection over the y-axis. Justify your reasoning. E(0, 1) E’(0, -1)
F(- 2 , 5) F’(2, 5)
G(-3, -4) G’(-3, 4)
H(5, 0) H’(-5, 0)
CHALLENGE Discuss the results of the following transformations. Suppose the figure is in Quadrant I. 30. Reflect a figure over the x-axis. Then reflect the image over the x-axis. 31. Reflect a figure over the x-axis. Then reflect the image over the y-axis. 32.
Writing in Math
How are transformations involved in recreational activities? Include an example of a recreational activity that represents each type of transformation. Lesson 10-3 Transformations on the Coordinate Plane
Pete Saloutos/CORBIS
529
33. Figure DEFG is dilated by a scale factor of _1 with the center at the origin. Which graph 3 shows this transformation?
Y £Ó ££ £ä ' n Ç È x { Î Ó £
n Ç È x { Î Ó £ "
B
C
Y
&g
$g
£ Ó Î { x È Ç nX
Y n Ç &g È x 'g { Î Ó £ "
%g
'g
"
D
%g $g
£ Ó Î { x È Ç nX
n Ç È x { Î Ó £
%
$
£ Ó Î { x È Ç n £äX
"
A
&
Y
&g %g
'g $g
£ Ó Î { x È Ç nX
Y n Ç È x &g { Î 'g %g Ó £ $g " £ Ó Î { x È Ç nX
Complete each congruence statement if ABC DEF. (Lesson 10-2) −− −− 34. ∠D ? 35. AC ? 36. DE ?
37. ∠C ?
38. ALGEBRA Angles A and B are complementary. If m∠A = (x + 3)° and m∠B is twice m∠A, write an equation that can be used to find the value of x. (Lesson 10-1) to estimate how many 39. SKYSCRAPERS Use the formula D = 1.22 × √A miles you can see from a point above the horizon. Suppose you are standing in the observation area of the Sears Tower in Chicago. About how far can you see on a clear day if the deck is 1353 feet above the ground? (Lesson 9-1) 40. Evaluate ⎪-4⎥ - ⎪3⎥. (Lesson 2-1)
PREREQUISITE SKILL Solve each equation. (Lesson 3-5) 41. 2x + 134 = 360
43. 5x + 125 = 360
42. 3x + 54 = 360
44. 4x + 92 = 360
530 Chapter 10 Two-Dimensional Figures
EXTEND
10-3
Geometry Lab
Rotations
Another type of transformation is a rotation. A rotation turns a figure with respect to a point called the center of rotation.
ACTIVITY 1 Step 1 Use grid paper to draw a triangle on a coordinate plane. Label the triangle ABC. Label the origin as O. Tape the coordinate plane to the desktop.
"g
/
#g
!g
Step 2 Trace the triangle onto a sheet of tracing paper. This will become the rotated image A’B’C’. Position the traced triangle over ABC on the grid paper. Then draw segment OA’ on the tracing paper.
Step 3 Place the point of your pencil through the hole of a protractor and −−− position it at the origin. Place the 0 line of the protractor on OA’. Step 4 Holding the protractor still, turn the tracing paper clockwise −−− until OA’ is at the 90° line.
ANALYZE THE RESULTS 1. How was ABC transformed to the image A’B’C’? 2. Copy and complete the tables using the coordinates of your triangles. 3. What do you observe about the coordinates of points A and A'?
x
y
x
A
A’
B
B’
C
C’
y
4. Repeat the Activity, rotating the figure 90° counterclockwise. Describe what you observe about the coordinates of points A and A'. 5. Repeat the Activity, rotating the figure 180° about the origin. What do you observe about the coordinates of points A and A’? 6. MAKE A CONJECTURE Write a rule to describe what happens to coordinates (x, y) of a figure after a clockwise rotation of 90°, a counterclockwise rotation of 90°, and a rotation of 180° about the origin. For Exercises 7–9, use the graph at the right.
y
7. Graph the image of the figure after a rotation of 90° counterclockwise. 8. Find the coordinates of the vertices of the figure after a 180° rotation. 9. Graph the image of the figure after a rotation of 90° clockwise.
B x
O A
C E
D
Extend 10-3 Geometry Lab: Rotations
531
10-4
Quadrilaterals
Main Ideas • Find the missing angle measures of a quadrilateral. • Classify quadrilaterals.
New Vocabulary quadrilateral
Geometric figures are often used to create various designs. Notice how the brick walkway at the right is formed using different-shaped bricks to create circles. a. Describe the bricks used to create the smallest circles. b. Describe how the shape of the bricks change as the circles get larger.
Quadrilaterals Squares, rectangles, and trapezoids are examples of quadrilaterals. A quadrilateral is a closed figure with four sides and four vertices. The segments of a quadrilateral intersect only at their endpoints. Quadrilaterals
Not Quadrilaterals
As with triangles, a quadrilateral can be named by its vertices. Two names of the quadrilateral below are quadrilateral ABCD and quadrilateral CBAD. Naming Quadrilaterals
The vertices are A, B, C, and D.
When you name a quadrilateral, you can begin at any vertex. However, it is important to name vertices in order.
The angles are A, B, C, and D.
A
B The sides are AB, BC, CD, and DA. C
D
A quadrilateral can be separated into two triangles. Since the sum of the measures of the angles of a triangle is 180°, the sum of the measures of the angles of a quadrilateral is 2(180°) or 360°. 532 Chapter 10 Two-Dimensional Figures Richard Hamilton Smith/CORBIS
F G E
H
Angles of a Quadrilateral Th e su m o f th e m e asu re s o f th e an gle s o f a q u ad rilate ral is 3 6 0 °.
EXAMPLE
Find Angle Measures
ALGEBRA Find the value of x. Then find each missing angle measure.
B 62˚ 2x ˚
The sum of the measures of the angles is 360°. Let m∠A, m∠B, m∠C, and m∠D represent the measures of the angles. m∠A + m∠B + m∠C + m∠D = 360
88 + 62 + 2x +
= 360
x
3x + 150 = 360 Check Your Work To check the answer, find the sum of the measures of the angles. Since 88° + 62° + 140° + 70° = 360°, the answer is correct.
A
88˚
C
x˚
D
Angles of a quadrilateral Substitution Combine like terms.
3x + 150 - 150 = 360 - 150 Subtract 150 from each side. 3x = 210 x = 70
Simplify. Divide each side by 3.
So, m∠D = 70° and m∠C = 2(70) or 140°.
1. ALGEBRA In quadrilateral EFGH, m∠E = 3x°, m∠F = 70°, m∠G = x°, and m∠H = 82°. Find the value of x. Then find each missing angle measure. Personal Tutor at pre-alg.com
Classify Quadrilaterals The diagram below shows how quadrilaterals are related. It goes from the most general quadrilateral to the most specific. The best description of a quadrilateral is the one that is the most specific. Animation pre-alg.com
+Õ>`À>ÌiÀ>
*>À>i}À> /À>«iâ` QUADRILATERAL WITH EXACTLY ONE PAIR OF PARALLEL SIDES
QUADRILATERAL WITH BOTH PAIRS OF OPPOSITE SIDES PARALLEL AND CONGRUENT
, LÕÃ PARALLELOGRAM WITH CONGRUENT SIDES
,iVÌ>}i PARALLELOGRAM WITH RIGHT ANGLES
Extra Examples at pre-alg.com
-µÕ>Ài PARALLELOGRAM WITH CONGRUENT SIDES AND RIGHT ANGLES
Lesson 10-4 Quadrilaterals
533
BASKETBALL The photograph shows the free-throw lane used during international competitions. Classify the quadrilateral using the name that best describes it. The quadrilateral has exactly one pair of opposite sides that are parallel. It is a trapezoid.
Classify each quadrilateral using the name that best describes it. 2A. 2B.
ALGEBRA Find the value of x. Then find the missing angle measures. 1.
x˚
68˚
125˚
2x ˚
118˚
x˚
Classify each quadrilateral using the name that best describes it. 3.
4.
HOMEWORK
HELP
For See Exercises Examples 6–11 1 12–17 2
8
10
ALGEBRA Find the value of x. Then find the missing angle measures. 6.
65˚
7. 109˚
128˚ 3x ˚ x˚
115˚
9.
8.
x˚ 96˚ 110˚
x˚
x˚
x˚
2x ˚
534 Chapter 10 Two-Dimensional Figures John Kolesidis/Reuters/CORBIS
8
5. SPORTS Classify the quadrilaterals that are found on the scoring region of a shuffleboard court.
10
(p. 534)
100˚
10
Example 2
2.
64˚
7
(p. 533)
7
Example 1
120˚
120˚
52˚
10.
2x ˚
x˚
11.
90˚ (2x 20)˚
135˚
(x 5)˚
x˚
(x 10)˚
Classify each quadrilateral using the name that best describes it. 12.
13.
14.
15.
16.
17.
18. ART Classify the quadrilaterals that are outlined in the painting at the right.
Real-World Career Artist An artist uses math to create paintings, sculptures, or illustrations to communicate ideas. For more information, go to pre-alg.com.
EXTRA
PRACTICE
See pages 784, 803. Self-Check Quiz at pre-alg.com
H.O.T. Problems
19. GAMES Identify a game that is played on a board that is shaped like a square. Describe the characteristics that make the board a square. 20. COOKING Name an item found in a kitchen that is rectangular in shape. Explain why the item is a rectangle. Determine whether each statement is sometimes, always, or never true. 21. A square is a rhombus. 22. A parallelogram is a rectangle. 23. A rectangle is a square. 24. A parallelogram is a quadrilateral.
Irene Rice Perlera. Untitled, 1951
Make a drawing of each quadrilateral. Then classify each quadrilateral using the name that best describes it. 25. In quadrilateral JKLM, m∠J = 90°, m∠K = 50°, m∠L = 90°, and m∠M = 130°. −−− −− −− −− 26. In quadrilateral CDEF, CD and EF are parallel, and CF and DE are parallel. Angle C is not congruent to ∠D. 27. ART The abstract painting at the right is an example of how shape and color are used in art. Write a few sentences describing the geometric shapes used by the artist. CHALLENGE For Exercises 28 and 29, use the following information. An equilateral figure is one in which all sides have the same measure. An equiangular figure is one in which all angles have the same measure. 28. Is it possible for a quadrilateral to be equilateral Elizabeth Murray Painter’s Progress, 1981 without being equiangular? If so, explain with a drawing. 29. Is it possible for a quadrilateral to be equiangular without being equilateral? If so, explain with a drawing. Lesson 10-4 Quadrilaterals
(tl)Pat LaCroix; (cr)“Untitled”, 1951. Irene Rica Pereira. Oil on board, 101.5 X 61 cm. Solomon R. Guggenheim Museum, New York, NY. Gift of Jerome B. Lurie, 1981.; (bl)“Painter’s Progress”, 1981. Elizabeth Murray. Museum of Modern Art, New York, NY. Aquired through the Bernhill Fund and gift of Agnes Gund/Art Resource, NY
535
30. OPEN ENDED Choose four cities on a map of the United States that when connected form a rectangle. How do you know it is a rectangle? 31.
Writing in Math How are quadrilaterals used in design? Include an example of a real-world design that contains quadrilaterals and an explanation of the figures used in the design.
32. Which figure is best described as a square? A C
B
33. GRIDDABLE Mrs. Smith used the parallelogram below to design a pattern for a paving stone. She will use the paving stone for a sidewalk. Find x.
D
Ýc
£Îäc
Ýc
£Îäc
34. A figure has vertices D(1, 2), E(1, 4), F(-4, 4), and G(-2, 2). Graph the figure and its image after a translation 4 units down. (Lesson 10-3) Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement. (Lesson 10-2) 35.
% £È vÌ
&
Èäc £Ó vÌ
xxc
Èxc £{ vÌ
36. ,
(
'
n vÌ
*
Èäc Èxc
È vÌ xxc
Ç vÌ
-
.
/
)
37. GARDENING Suppose you plant a square garden with an area of 300 square feet. How much fencing will you need to buy to enclose the garden if the fencing only comes in whole-foot sections? (Lesson 9-2) 38. ALGEBRA Solve x - 3.4 ≥ 6.2. Graph the solution on a number line. (Lesson 8-4) 39. GEOGRAPHY Forty-six percent of the world’s water is in the Pacific Ocean. What fraction is this? (Lesson 6-5) 40. TRACK AND FIELD Heather needs to average 11.4 seconds in the 100-meter dash in six races to qualify for the championship race. The mean of her first five races was 11.2 seconds. What is the greatest time that she can run and still qualify for the race? (Lesson 5-9)
PREREQUISITE SKILL Simplify each expression. (Lesson 1-2) 41. (5 - 2)180 42. (7 - 2)180 43. (10 - 2)180 536 Chapter 10 Two-Dimensional Figures
44. (9 - 2)180
CH
APTER
10
Mid-Chapter Quiz Lessons 10-1 through 10-4
1. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 60° angle with the post as shown. Find y. (Lesson 10-1)
10. MULTIPLE CHOICE Triangle JKL has vertices J(3, 5), K(5, 7), and L(6, 3) and is dilated by a scale factor of 1 with the origin as 3 the center of dilation. What are the coordinates of L’? (Lesson 10-3)
Èäc
Þc
A (2, 1)
B 1, 1 3 C (18, 9)
2. If m∠Y = 23° and ∠Y and ∠Z are complementary, what is m∠Z? (Lesson 10-1)
D 3, 3
3. Angles G and H are supplementary. If m∠G = x + 11 and m∠H = x - 13, what is the measure of each angle? (Lesson 10-1) In the figure m and t is a transversal. Find the value of x for each of the following. (Lesson 10 -1)
11. After a translation of 4 units left and 2 units up, the coordinates of the vertices of the image ABC are A(3, 2), B(0, 4), and C(-3, 5). What were the coordinates of the vertices before the translation? (Lesson 10 -3)
t
m n
x Ç
£ Ó { Î È
Find the value of x. Then find the missing angle measures. (Lesson 10 - 4)
4. m∠3 = 3x + 9 and m∠5 = 6x + 12 5. m∠6 = 8x + 7 and m∠8 = 9x - 10 6. UMBRELLAS An umbrella has eight ) congruent triangular sections with spokes of equal length. " Name one pair of congruent triangles. Then find the # corresponding parts. (Lesson 10-2)
2
12.
(
25˚
'
175˚
x˚
13.
95˚
!
2x ˚
x˚
&
PARKING SPACES Classify each quadrilateral using the name that best describes it. (Lesson 10 - 4) %
14.
15.
$
7. Suppose ABC DEF. Which angle is congruent to ∠D? (Lesson 10-2) 8. Triangle QRS has vertices Q(3, 3), R(5, 6), and S(7, 3). Find the coordinates of the vertices after the triangle is reflected over the x-axis.
16. MULTIPLE CHOICE Which figure is best described as a parallelogram? (Lesson 10-4) F
H
G
J
(Lesson 10-3)
9. GAMES What type of transformation is used when a checker piece is moved on a checkerboard? (Lesson 10-3)
Chapter 10 Mid-Chapter Quiz
537
Learning Mathematics Prefixes Quadruplets are four children born at the same tim to the same mother. The prefix quad- also appears in the term quadrilateral—a polygon with four sides. The table shows some of the prefixes that are used in mathematics. These prefixes are also used in everyday language. In order to use each prefix correctly, you need to understand its meaning.
Prefix
Meaning
Everyday Words
four
quadruple quadruplet quadriceps
a sum four times as great as another one of four offspring born at one birth a muscle with four points of origin
Pentagon pentathlon pentad
headquarters of the Department of Defense a five-event athletic contest a group of five
quad-
Meaning
pent-
five
hex-
six
hexapod hexagonal hexastich
having six feet having six sides a poem of six lines
hept-
seven
hepted heptagonal heptarchy
a group of seven having seven sides a government by seven rulers
oct-
eight
octopus octet octennial
a type of mollusk having eight arms a musical composition for eight instruments lasting eight years
dec-
ten
decade decameter decathlon
a period of ten years ten meters a ten-event athletic contest
Reading to Learn 1. Refer to the table above. For each prefix listed, choose one of the everyday words listed and write a sentence that contains the word. 2. RESEARCH Use the Internet, a dictionary, or another reference source to find a mathematical term that contains each of the prefixes listed. Write the definition of each term. 3. RESEARCH Use the Internet, a dictionary, or another reference source to find a different word that contains each prefix. Then define the term. 538 Chapter 10 Two-Dimensional Figures Ace Stock Limited/Alamy Images
10-5
Polygons
Main Ideas • Classify polygons. • Determine the sum of the measures of the interior and exterior angles of a polygon.
New Vocabulary polygon diagonal interior angles regular polygon
The tiled patterns below are called regular tessellations. Notice how the figures repeat to form patterns that contain no gaps or overlaps.
Square Tessellation
Triangle Tessellation Hexagon Tessellation
a. Which figure is used to create each tessellation?
vertex
b. Refer to the diagram at the right. What is the sum of the measures of the angles that surround the vertex? c. Does the sum in part b hold true for the square tessellation? Explain. d. Make a conjecture about the sum of the measures of the angles that surround a vertex in the hexagon tessellation.
Classify Polygons A polygon consists of a sequence of consecutive line segments placed end to end to form a simple closed figure. The figures below are examples of polygons. The line segments meet only at their endpoints.
The points of intersection are called vertices.
The line segments are called sides.
The following figures are not polygons.
This is not a polygon because it has a curved side. n-gon A polygon with n sides is called an n-gon. For example, an octagon can also be called an 8-gon.
This is not a polygon because it is an open fugure.
This is not a polygon because the sides overlap.
Polygons can be classified by the number of sides they have. Name of Polygon
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Number of Sides
5
6
7
8
9
10
Lesson 10-5 Polygons
539
EXAMPLE
Classify Polygons
Classify each Polygon. a.
b.
The polygon has 8 sides. It is an octagon.
1A.
The polygon has 6 sides. It is a hexagon.
1B.
Measures of the Angles of a Polygon A diagonal is a line segment in a polygon that joins two nonconsecutive vertices. In the diagram below, all possible diagonals from one vertex are shown. quadrilateral
pentagon
hexagon
heptagon
octagon
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Sides
4
5
6
7
8
Diagonals
1
2
3
4
5
Triangles
2
3
4
5
6
Notice that the number of triangles is 2 less than the number of sides.
You can use the property of the sum of the measures of the angles of a triangle to find the sum of the measures of the interior angles of any polygon. An interior angle is an angle inside a polygon. Interior Angles of a Polygon If a polygon has n sides, then n - 2 triangles are formed. The sum of the degree measures of the interior angles of the polygon is (n - 2)180.
EXAMPLE
Measures of Interior Angles
Find the sum of the measures of the interior angles of a heptagon. A heptagon has 7 sides. Therefore, n = 7. (n - 2)180 = (7 - 2)180 Replace n with 7. = 5(180) or 900° Simplify.
2. Find the sum of the measures of the interior angles of a 13-gon. 540 Chapter 10 Two-Dimensional Figures
Extra Examples at pre-alg.com
A regular polygon is a polygon that is equilateral (all sides are congruent) and equiangular (all angles are congruent). Since the angles of a regular polygon are congruent, their measures are equal.
SNOW Snowflakes are some of the most beautiful objects in nature. Notice how they are regular and hexagonal in shape. What is the measure of one interior angle in a snowflake? Step 1 Find the sum of the measures of the angles. A hexagon has 6 sides. Therefore, n = 6. Real-World Link Snowflakes are also called snow crystals. It is said that no two snowflakes are alike. They differ from each other in size, lacy structure, and surface markings.
(n - 2)180 = (6 - 2)180 Replace n with 6. = 4(180) or 720 Simplify. The sum of the measures of the interior angles is 720°. Step 2 Divide the sum by 6 to find the measure of one angle. 720 ÷ 6 = 120 So, the measure of one interior angle in a snowflake is 120°.
Source: infoplease.com
3. KALEIDOSCOPE This kaleidoscope is regular and nonagonal in shape. What is the measure of one interior angle of the nonagon?
Personal Tutor at pre-alg.com
Example 1 (p. 540)
Classify each polygon. Then determine whether it appears to be regular or not regular. 1.
2.
3. TESSELLATIONS Identify the polygons that are used to create the tessellation shown at the right. Example 2 (p. 540)
Example 3 (p. 541)
4. Find the sum of the measures of the interior angles of a nonagon. 5. What is the measure of each interior angle of a regular heptagon? Round to the nearest tenth. Lesson 10-5 Polygons
(l)Steve Austin/Papilio/CORBIS; (r)Becky Hayes/age fotostock
541
HOMEWORK
HELP
For See Exercises Examples 6–11 1 12–17 2 18–23 3
Classify each polygon. Then determine whether it appears to be regular or not regular. 6.
7.
8.
9.
10.
11.
Find the sum of the measures of the interior angles of each polygon. 12. pentagon 15. hexagon
13. octagon 16. 18-gon
14. decagon 17. 23-gon
Find the measure of an interior angle of each polygon. 18. regular nonagon 21. regular decagon
19. regular pentagon 22. regular 12-gon
20. regular octagon 23. regular 25-gon
ART For Exercises 24 and 25, use the painting below.
Roy Lichtenstein. Modern Painting with Clef. 1967
24. List five polygons used in the painting. 25. RESEARCH The title of the painting mentions the music symbol, clef. Use the Internet or another source to find a drawing of a clef. Is a clef a polygon? Explain. TESSELLATIONS For Exercises 26 and 27, identify the polygons used to create each tessellation. 27. 26.
EXTRA
PRACTICE
See pages 784, 803. Self-Check Quiz at pre-alg.com
28. BASEBALL The star in the Houston Astros’ logo is regular and pentagonal in shape. What is the measure of one interion angle in the pentagon? 29. ART Refer to Exercise 3 on page 541. The tessellation design contains regular polygons. Find the perimeter of the design if the measure of the sides of the 12-gon is 5 centimeters.
542 Chapter 10 Two-Dimensional Figures (t)“Modern Painting with Clef”, 1967. Roy Lichtenstein. Oil on synthetic polymer and pencil on canvas, 252 x 458 cm. Hirschhorn Museum and Sculpture Garden, Smithsonian Institution, Washington D.C.; (b)Courtesy of Houston Astros & Major League Baseball
H.O.T. Problems
30. SELECT A TOOL Study the dot pattern shown at the right. Which of the following tools would you use to find four line segments connecting all of the points that were drawn without lifting a pencil from the paper? Justify your selection. Then use the tool to solve the problem. draw a model
paper/pencil
technology
31. OPEN ENDED Draw a polygon that is both equiangular and equilateral. CHALLENGE When a side of a polygon is extended, an exterior angle is formed. In any polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.
72˚
35.
33. regular triangle
exterior angle
72˚
Find the measure of each exterior angle of each regular polygon. 32. regular octagon
72˚
72˚ 72˚
34. regular decagon
Writing in Math
How are polygons used in tessellations? Draw an example of a tessellation in which the pattern is formed using only one type of polygon and an example of a tessellation in which the pattern is formed using more than one polygon.
36. A landscape architect is looking for a brick paver shape that will tessellate. Which shape will allow her to tessellate a patio area? A
C
B
D
37. Which term identifies the shaded part of the design shown?
F heptagon
H octagon
G hexagon
J pentagon
Classify each quadrilateral using the name that best describes it. (Lesson 10-4) 38.
39.
40.
41. Triangle MNP has vertices M(-1, 1), N(5, 4), and P(4, 1). Graph the triangle after a translation 3 units left and 4 units down. (Lesson 10-3) 42. ALGEBRA Simplify 4.6x + 2.5x + 9.3x. (Lesson 3-2)
PREREQUISITE SKILL Find each product. (pages 747–748) 43. (3)(4.8) 44. (5.4)(6) 45. (9.2)(3.1)
46. (10.5)(5.7)
Lesson 10-5 Polygons
543
Geometry Lab
EXTEND
10-5
Tessellations
A tessellation is a pattern of repeating figures that fit together with no overlapping or empty spaces. Tessellations can be formed using transformations.
ACTIVITY 1 Create a tessellation using a translation. Step 1 Draw a square. Then draw a triangle inside the top of the square as shown.
Step 3 Repeat this pattern unit to create a tessellation. It is sometimes helpful to complete one pattern, cut it out, and trace it for the other pattern units.
Step 2 Translate or slide the triangle from the top to the bottom of the square. Step 1
Step 2
ACTIVITY 2 Create a tessellation using a rotation. Step 1 Draw an equilateral triangle. Then draw another triangle inside the left side of the triangle as shown below.
Step 3 Repeat this pattern unit to create a tessellation.
Step 2 Rotate the triangle so you can trace the change on the side as indicated. Step 1
Step 2
EXERCISES Use a translation to create a tessellation for each pattern unit shown. 1.
2.
3.
Use a rotation to create a tessellation for each pattern unit shown. 4.
5.
6.
7. Make a tessellation that involves a translation, a rotation, or a combination of the two. 544 Chapter 10 Two-Dimensional Figures
10-6
Area: Parallelograms, Triangles, and Trapezoids
Main Ideas • Find areas of parallelograms. • Find the areas of triangles and trapezoids.
The area of a rectangle can be found by multiplying the length and width. The rectangle shown below has an area of 3 6 or 18 square units. Suppose a triangle is cut from one side of the rectangle and moved to the other side. The new figure is a parallelogram.
New Vocabulary base altitude
UNITS UNITS
a. Compare the area of the rectangle to the area of the parallelogram. b. What parts of a rectangle and parallelogram determine their area?
Areas of Parallelograms The area of a parallelogram can be found by multiplying the measures of the base and the height.
TThe e base can a be e an side any e of the e p a eo parallelogram.
The e height he g is the length eng of an n altitude, a line segment eg p rp perpendicular c a to the e base as with w e po endpoints on o the e base a e and n the side de opposite pp the base.
altitude base
Area of a Parallelogram Words
If a parallelogram has a base of b units and a height of h units, then the area A is bh square units.
Model
b h
Symbols A = bh
EXAMPLE Generating Formulas For a lesson on generating formulas involving area, see page 807.
Find Areas of Parallelograms
Find the area of each parallelogram. a.
The base is 14 feet. The height is 12 feet. 12 ft
Estimate A = 15 · 10 or 150 14 ft
A = bh
Area of a parallelogram
= 14 • 12 Replace b with 14 and h with 12. = 168
Multiply.
The area is 168 square feet. The answer is close to the estimate so the answer is reasonable. Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids
545
b.
The base is 5.9 centimeters. The height is 7.5 centimeters. Estimate A = 6 8 or 48
Altitudes
7.5 cm
An altitude can be outside the parallelogram.
A = bh
Area of a parallelogram
= (5.9)(7.5) Replace b with 5.9 and h with 7.5. = 44.25
5.9 cm
Multiply.
The area is 44.25 square centimeters. Is the answer reasonable? Find the area of each parallelogram. 1A.
1B. Î
°Ç
È £ä°Î
Area of Triangles and Trapezoids A diagonal of a parallelogram separates the parallelogram into two congruent triangles. The area of each triangle is one-half the area of the parallelogram. diagonal
The area of parallelogram ABCD is 7 ⭈ 4 or 28 square units.
B
C 4
A
The area of triangle ABD is 21 ⭈ 28 or 14 square units.
D
7
Using the formula for the area of a parallelogram, we can find the formula for the area of a triangle. Area of a Triangle If a triangle has a base of b units and a height of h units, then the 1 area A is _ bh square units.
Words
1 Symbols A = _ bh
Model h
2
b
2
EXAMPLE Alternative Method Multiplication is commutative and associative. So you can also find 12 of 6 first and then multiply by 5.
Find Areas of Triangles
Find the area of each triangle. a.
The base is 5 inches. The height is 6 inches.
b.
The base is 7 meters. The height is 4.2 meters.
4.2 m
6 in. 7m
1 A=_ bh
5 in.
1 A=_ bh
Area of a triangle 2 1 =_ (5)(6) Replace b with 5 and h with 6. 2 1 =_ (30) Multiply. 5 × 6 = 30 2
= 15
in2
546 Chapter 10 Two-Dimensional Figures
Simplify.
2 1 _ = (7)(4.2) 2 1 =_ (29.4) 2
= 14.7 m2
Area of a triangle Replace b with 7 and h with 4.2. Multiply 7 × 4.2 = 29.4 Simplify.
2A.
2B.
Î vÌ
Ó V £ Î
x vÌ
£ä V
A trapezoid has two bases. The height of a trapezoid is the distance between the bases. A trapezoid can be separated into two triangles. F
G
a
K
h
E
&
h
%
A
+
'
H
H
H
b
J
&
(
B
(
area of trapezoid EFGH = area of EFH + area of FGH
_1 bh
=
1 =_ h(a + b) 2
_1 ah
+
2
2
Distributive Property
Area of a Trapezoid Words
If a trapezoid has bases of a units and b units and a height of h units, then the area A of 1 the trapezoid is _ h(a + b) 2 square units.
a
Model h
b
1 Symbols A = _ h(a + b) 2
EXAMPLE Look Back To review multiplying fractions, see Lesson 5-3.
Find Area of a Trapezoid
Find the area of the trapezoid. 1 The height is 4 inches. The bases are 6 _ inches 2 1 _ and 3 inches. 4 Estimate 12(4)(7 + 3) or 20 1 A=_ h(a + b) 2 1 =_ · 4 61 + 31 2 4 2 3 1 =_ · 4 · 9_ 4 2 39 1 _ 4 _ _ = · · 2 1 4 39 = or 19 1 in2 2 2
( _ _)
Area of a trapezoid
6 1 in. 2
4 in.
3 1 in. 4
Replace h with 4, a with 6 12 and b with 314. 6 12 + 314 = 9 34 Divide out the common factors.
1 3. Find the area of a trapezoid with a height of 8_ yards and bases that 4 1 _ are 5 yards and 3 yards long. 3
Extra Examples at pre-alg.com
Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids
547
FLAGS The signal flag shown represents the number five. Find the area of the blue region. Estimate The blue region is about
_1 of the whole flag.
IN
1 1 So, A = _bh = _(14)(14) or about 50. 4
4
IN
IN
4
To find the area of the blue region, subtract the areas of the triangles from the area of the square. Area of the square
IN
Area of each triangle 1 A=_ bh
A = bh = 14 · 14 = 196
2
1 = _ · 12 · 6 2 = 36
The total area of the triangles is 4(36) or 144 square inches. So, the area of the blue region is 196 - 144 or 52 square inches. The answer is close to the estimate so the answer is reasonable.
4. FLAGS The flag shown below is the international signal for the number three. Find the area of the red region. 18 in.
9 in.
3 in.
7 in. 12 in.
Personal Tutor at pre-alg.com
Examples 1–3 (pp. 545–547)
Find the area of each figure shown or described. 1.
2. 4 ft
3. 5.4 cm
2 ft
15 m 6m 8m
3 cm
7.6 m 5m
4. parallelogram: base, 5.6 m; height, 9.4 m 5. triangle: base, 12 km; height 13 km 6. trapezoid: height: 16 in.; bases, 3.1 in. and 7.6 in. Example 4 (p. 548)
7. Find the approximate area of the state of Nevada. Round to the nearest square mile.
MI MI .EVADA
548 Chapter 10 Two-Dimensional Figures
MI
HOMEWORK
HELP
For See Exercises Examples 8–19 1–3 20, 21 4
Find the area of each figure. 8.
9.
Ç°{ °
10.
15 cm
ΰx °
11.
12. 10 in.
12 cm
5.5 m
2m
13.
6.7 cm
5.3 cm
7.2 in.
Ó{ vÌ £ ££ vÌ x
£Ó vÌ
9.9 cm
£{ vÌ
6 in.
Find the area of each figure described. 14. triangle: base, 8 in.; height, 7 in. 15. trapezoid: height, 2 cm; bases, 3 cm, 6 cm 16. parallelogram: base, 3.8 yd; height, 6 yd 17. triangle: base, 9 ft; height, 3.2 ft 18. trapezoid: height, 3.5 m; bases, 10 m and 11 m 19. parallelogram: base, 5.6 km; height, 4.5 km GEOGRAPHY For Exercises 20 and 21, use the approximate measurements to estimate the area of each state. 20.
287 mi
Real-World Link The largest U.S. state is Alaska. It has an area of 615,230 square miles. Rhode Island is the smallest state. It has an area of 1231 square miles. Source: The World Almanac
21.
332 mi
270 mi
ARKANSAS
OREGON
235 mi
165 mi
22. Find the base of a parallelogram with a height of 9.2 meters and an area of 36.8 square meters. 23. Suppose a triangle has an area of 20 square inches and a base of 2 1 inches. 2 What is the measure of the height? 24. A trapezoid has an area of 54 square feet. What is the measure of the height if the bases measure 16 feet and 8 feet? Find the area of each figure with the vertices shown. 25. rectangle: A(-3, 4), B(5, 4), C(5, -1), D(-3, -1) 26. triangle: E(-2.5, 2), F(3, -2.5), G(3, 2) Find the area of each figure. 27.
EXTRA
28.
12 km
PRACTICE
2 km
3m
4m
8 km 15 km
8 ft
9m
8m
See pages 785, 803. Self-Check Quiz at pre-alg.com
29. 6 ft
6m 8 ft 11 ft
Lesson 10-6 Area: Parallelograms, Triangles, and Trapezoids Lewis Kemper/SuperStock
549
30. LAWNCARE Mrs. Malone plans to fertilize her lawn. The fertilizer she will be using indicates that one bag fertilizes 2000 square feet. How many bags of fertilizer should she buy?
H.O.T. Problems
15 ft
125 ft Lawn 15 ft
Flower bed
84 ft
Patio 12 ft
31. OPEN ENDED Draw and label a parallelogram that has an area of 24 square inches.
100 ft
32. CHALLENGE Explain how the formula for the area of a trapezoid can be used to find the formulas for the areas of parallelograms and triangles. 33.
Writing in Math How is the area of a parallelogram related to the area of a rectangle? Describe the similarities and differences between a rectangle and a parallelogram, and draw a diagram that shows how the area of a parallelogram is related to the area of a rectangle.
34. The developer of a park wants to 320 ft change the base 160 ft measures but keep 560 ft the park the same size. Which base measures are NOT possible if the height stays the same? A 220 ft, 660 ft
C 231 ft, 649 ft
B 353 ft, 527 ft
D 370 ft, 610 ft
35. Which figure does NOT have an area of 48 square meters? F
H
8m
3m 1.6 m
6m
G
J
8m
12.8 m 5m
12 m 6.4 m
Find the measure of an interior angle of each polygon. (Lesson 10-5) 36. regular hexagon
37. regular decagon 38. regular octagon
39. regular 15-gon
Find the value of x. Then find the missing angle measures. (Lesson 10-4) 40.
60˚
x˚ 60˚
120˚
41. 110˚ 4x ˚
130˚
42. TAXI FARES The table shows taxi ride fares for two companies. Compare the rates of change. (Lesson 7-5)
x˚
Miles (min) x 0 3 9
Fare y Taxi 1 Taxi 2 $0 $0 $2.40 $2.70 $7.20 $8.10
PREREQUISITE SKILL Use a calculator to find each product. Round to the nearest tenth. (pages 747–748) 43. 3.14 · 4.3 44. 2 · 3.14 · 5.4 45. 3.14 · 42 46. 3.14(2.4)2 550 Chapter 10 Two-Dimensional Figures
10-7
Circles: Circumference and Area
Main Ideas • Find circumference of circles. • Find area of circles.
New Vocabulary circle diameter center circumference radius (pi)
Coins, paper plates, cookies, and CDs are all examples of objects that are circular in shape. d
Object
a. Collect three different-sized circular objects. Then copy the table shown.
C
C d
1 2 3
b. Using a tape measure, measure each distance below to the nearest millimeter. Record your results. • the distance across the circular object through its center (d) • the distance around each circular object (C) C c. For each object, find the ratio _ . Record the results in the table. d
d. Write an equation that relates the circumference C of a circle to its diameter d.
Circumference of Circles A circle is the set of all points in a plane that are the same distance from a given point.
Pi Although π is an irrational number, 3.14 22 and _ are two 7 generally accepted approximations for π.
The distance across the circle through its center is its diameter.
The given point is called the center.
The distance around the circle is called the circumference.
The distance from the center to any point on the circle is its radius.
The relationship you discovered above is true for all circles. The ratio of the circumference of a circle to its diameter is always equal to 3.1415926… The Greek letter π (pi) stands for this number. Using this ratio, you can derive a formula for the circumference of a circle. C _ =π
d C _ ·d =π·d d
C = πd
The ratio of the circumference to the diameter equals pi. Multiply each side by d. Simplify.
Circumference of a Circle Words
The circumference of a circle is equal to its diameter times π, or 2 times its radius times π.
Model C d
r
Symbols C = πd or C = 2πr Lesson 10-7 Circles: Circumference and Area
551
Interactive Lab pre-alg.com
If an exact answer is required, leave the answer in terms of . A decimal 22 can be used for estimating answers. approximation of , 3.14, or _ 7
EXAMPLE
Find the Circumference of a Circle
Find the circumference of each circle to the nearest tenth. C = d
a. 5 cm
Circumference of a circle
=·5
Replace d with 5.
= 5
Simplify. This is the exact circumference.
To approximate the circumference, first use ≈ 3 to get an estimate. Then use a calculator. Estimate: 5 × ≈ 5 × 3 or about 15 Calculating with π Unless otherwise specified, use a calculator to evaluate expressions involving and then follow any instructions regarding rounding.
5 × 2nd [] ENTER 15.70796327 The circumference is about 15.7 centimeters. b.
C = 2r
3.2 ft
Circumference of a circle
= 2 · · 3.2
Replace r with 3.2.
= 20.1
Simplify. Use a calculator.
3 1A. diameter = 3_ ft
1B. radius = 7 mm
4
Personal Tutor at pre-alg.com
TREES A tree in Madison’s yard was damaged in a storm. Her parents want to replace the tree with another whose trunk is the same size as the original tree. Suppose the circumference of the original tree was 14 inches. What should be the diameter of the replacement tree? Explore You know the circumference of the original tree. You need to find the diameter of the new tree. Plan Solve
Use the formula for the circumference of a circle to find the diameter. C = d
Circumference of a circle
14 = · d Replace C with 14. 14 _ =d
Divide each side by .
4.5 ≈ d
Simplify. Use a calculator.
The diameter of the tree should be about 4.5 inches. Check
Is the solution reasonable? Check by replacing d with 4.5 in C = d. C = d
Circumference of a circle
= · 4.5 Replace d with 4.5. ≈ 14.1
Simplify. Use a calculator. The solution is reasonable.
2. MUSIC A CD has a diameter of 120 millimeters. A Universal Media Disc (UMD) has a diameter of 60 millimeters. Compare the circumferences of both disc sizes. 552 Chapter 10 Two-Dimensional Figures
Extra Examples at pre-alg.com
Areas of Circles A circle can be separated into parts as shown below. The parts can then be arranged to form a figure that resembles a parallelogram. #IRCUMFERENCE 2ADIUS #IRCUMFERENCE
Review Vocabulary
Since the circle has an area that is relatively close to the area of the figure, you can use the formula for the area of a parallelogram to find the area of a circle.
Exponents in a power, the number of times the base is used as a factor; Example: 53; 3 is the exponent (Lesson 4-1)
A = bh
Area of a parallelogram
1 A= _ × C r
The base of the parallelogram is one-half the circumference, and the radius is the height.
A = 1 × 2r r 2
Replace C with 2r.
2
A=×r×r
Simplify.
A = r2
Replace r × r with r2.
Area of a Circle Words
The area of a circle is equal to times the square of its radius.
Model r
Symbols A = r2
EXAMPLE
Find Areas of Circles
Estimation
Find the area of each circle. Round to the nearest tenth.
To estimate the area of a circle, square the radius and then multiply by 3.
a.
Estimate 3 · 36 or 108 6 in.
b.
A = r2 =·
Area of a circle
62
Replace r with 6.
= · 36
Evaluate 62.
≈ 113.1 in2
Use a calculator. The answer is reasonable.
Estimate 3 · 256 or 768
A = r2
31 m
Area of a circle
= · 31 2
3A.
2
Since d is 31, r is _. 31 2
= · (15.5)2
31 _ = 15.5
= · 240.25
Evaluate (15.5)2.
≈ 754.8 m2
Use a calculator. The answer is reasonable.
2
3B. £
£Î Ó ° ££
Lesson 10-7 Circles: Circumference and Area
553
Examples 1, 3 (pp. 552, 553)
Find the circumference and area of each circle. Round to the nearest tenth. 1.
2.
3.
5 mi
4 in. 8m
4. The radius is 1.3 kilometers. Example 2 (p. 552)
HOMEWORK
HELP
For See Exercises Examples 7–16 1, 3 17, 18 2
5. The diameter is 6.1 centimeters.
6. MUSIC During a football game, the marching band can be heard within a radius of 1.7 miles. What is the area of the neighborhood that can hear the band?
Find the circumference and area of each circle. Round to the nearest tenth. 7.
8.
9. 13 in.
6 cm
10 m
10.
11.
12.
21 km
13. The radius is 4.5 meters. 4 feet. 15. The diameter is 7 _ 5
1
9 2 ft
12.7 m
14. The diameter is 7.3 centimeters. 3 16. The radius is 15 _ inches. 8
17. LAWN CARE Sunki has a sprinkler positioned in her lawn that directs a 12-foot spray in a circular pattern. About how much of the lawn does the sprinkler water? 18. SCIENCE The circumference of the Moon is about 6790 miles. What is the distance to the center of the Moon?
Real-World Link The world’s largest fountain is the Suntec City Fountain of Wealth in Singapore. Made of cast bronze, it covers a total area of 18,117 square feet. Source: guinnessworldrecords.com
Match each circle described in the column on the left with its corresponding measurement in the column on the right. 19. radius: 4 units a. circumference: 37.7 units 20. diameter: 7 units b. area: 7.1 units2 21. diameter: 3 units c. area: 50.3 units2 22. radius: 6 units d. circumference: 22.0 units 23. What is the diameter of a circle if its circumference is 25.8 inches? Round to the nearest tenth. 24. Find the radius of a circle if its area is 254.5 square inches. 25. CAROUSEL The Carousel in Spring Green, Wisconsin is the world’s largest carousel. If it has a diameter of 80 feet, what is the distance a seat on it travels in 10 revolutions? Round to the nearest foot. 26. FOUNTAINS A circular fountain at a park has a radius of 4 feet. The mayor wants to build a fountain that is quadruple the size of the current fountain. Find the length of the radius of the new fountain.
554 Chapter 10 Two-Dimensional Figures eye35.com/Alamy Images
MI
Ü iÀV>à >Ì > ««i ANALYZE GRAPHS For Exercises 27–29, the circle graph at the right has a radius of 1 inch. Suppose the circle graph is {ǯ redrawn onto a poster board so Ìi Ì Ì that the diameter is tripled. ί ½Ì Ü 27. How much space on the poster board will the circle graph ££¯ *ii Ì Î¯ cover?
ÕÌ Ì Ì 28. How much of the total space ÃVià will each section of the graph cover? -ÕÀVi\ "« ,iÃi>ÀV vÀ -«i`> 29. How many times greater is the area of the new graph than the area of the original graph at the right? Is there, if any, a relationship with the increase in diameter and increase in area from one graph to the other? Explain.
Find the distance around and area of each figure. Round to the nearest tenth. Look Back
30. semicircle
31. semicircle
To review slope, see Lesson 7-5.
32. quarter circle
£ä vÌ x °
n
EXTRA
PRACTICE
See pages 785, 803. Self-Check Quiz at pre-alg.com
H.O.T. Problems
33. FUNCTIONS Graph the circumference of a circle as a function of the diameter. Use values of d like 1, 2, 3, 4, and so on. What is the slope of this graph? How is the slope related to the formula for finding circumference? 34. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose some data and write a real-world problem in which you would find the circumference or area of a circle. 35. OPEN ENDED Draw and label a circle that has an area between 5 and 8 square units. 36. FIND THE ERROR Dario and Peter are finding the area of a circle with a diameter of 7. Who is correct? Explain your reasoning. Dario A = πr2 = π(7)2 ≈ 153.9 units2
Peter A = πr2 = π(3.5)2 ≈ 38.5 units2
37. CHALLENGE The radius of circle B is 2.5 times the radius of circle A. If the area of circle A is 8 square yards, what is the area of circle B?
!
"
Lesson 10-7 Circles: Circumference and Area
555
38. NUMBER SENSE The numerical value of the area of a circle is twice the numerical value of the circumference. What is the radius of the circle? (Hint: Use a table of values for radius, circumference, and area.) 39.
Writing in Math How are circumference and diameter related? Give the ratio of the circumference to the diameter and describe what happens to the circumference as the diameter increases or decreases.
40. The Blackwells have a circular pool with a radius of 10 feet. They plan on installing a 3-foot-wide walkway around the pool. What will be the area of the walkway?
41. A sprinkler is set to cover the area shown. Find the area of the grass being watered if the sprinkler reaches a distance of 20 feet.
3 ft 20 ft 10 ft
A 216.8 ft2
C 314.2 ft2
B 285.9 ft2
D 442.2 ft2
F 78.5 ft2
H 942.5 ft2
G 314.2 ft2
J 1,256.6 ft2
Find the area of each figure described. (Lesson 10-6) 42. trapezoid: height, 2 m; bases, 20 m and 18 m 43. parallelogram: base, 6 km; height, 8 km Find the sum of the measures of the interior angles of each polygon. (Lesson 10-5) 44. pentagon
45. quadrilateral
46. octagon
47. ALGEBRA Solve 2x - 7 > 5x + 14. (Lesson 8-6) 48. INTERNET SHOPPING For every order submitted, an online bookstore charges a $5 shipping fee plus a charge on the weight of the items being shipped of $2 per pound. The total shipping charges y can be represented by y = 2x + 5, where x represents the weight of the order in pounds. Graph the equation. (Lesson 7-6) 49. TRAVEL Jessica’s flight to Chicago leaves Rome, Italy at 4:30 P.M. on Tuesday. The flight time is 8.5 hours. If Rome is 7 hours ahead of Chicago, use Chicago time to determine when she is scheduled to arrive. (Lesson 1-1)
PREREQUISITE SKILL Find each sum. (page 745) 50. 200 + 43.9 51. 23.6 + 126.9 556 Chapter 10 Two-Dimensional Figures
52. 345.14 + 23.8
53. 720.16 + 54.7
EXTEND
10-7
Spreadsheet Lab
Circle Graphs and Spreadsheets
In the following example , you will learn how to use a computer spreadsheet program to graph the results of a probability experiment in a circle graph.
EXAMPLE A spinner like the one shown at the right was spun 20 times each for two trials. The data are shown below. Use a spreadsheet to make a circle graph of the result. Step 1 Enter the data in a spreadsheet as shown.
Circle Graphs.xls B C D E A Blue Red Yellow 1 10 17 13 2 Total Trials 4 8 8 3 Trial 1 6 9 5 4 Trial 2 5 Total Trials 6 7 8 9 10 Blue 11 Red 12 Yellow 13 14 15 Sheet 1
Sheet 2
The spreadsheet evaluates the formula ⫽SUM(D3:D4) to find the total.
Sheet 3
Step 2 Select the data to be included in your graph. Then use the graph tool to create the graph. The spreadsheet will allow you to add titles, change colors, and so on.
EXERCISES 1. Describe the results you would theoretically expect for one trial of 20 spins. Explain your reasoning. 2. Make a spinner like the one shown above. Collect data for five trials of 20 spins each. Use a spreadsheet program to create a circle graph of the data. 3. A central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle. After 100 spins, what kind of central angle would you theoretically expect for each section of the circle graph? Explain. 4. Predict how many trials of the experiment are required to match the theoretical results. Test your prediction. 5. When the theoretical results match the experimental results, what is true about the circle graph and the spinner? Extend 10-7 Spreadsheet Lab: Circle Graphs and Spreadsheets
557
10-8
Area: Composite Figures
Main Idea • Find area of composite figures.
New Vocabulary composite figures
California is the most populous state in the United States. It ranks third among the U.S.