Unformatted Attachment Preview
7
TH
EDITION
ALGEBRA FOR
COLLEGE STUDENTS
This page intentionally left blank
7
TH
EDITION
ALGEBRA FOR
COLLEGE STUDENTS
Margaret L. Lial
American River College
John Hornsby
University of New Orleans
Terry McGinnis
Addison-Wesley
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Editorial Director: Christine Hoag
Editor-in-Chief: Maureen O’Connor
Executive Content Manager: Kari Heen
Content Editor: Courtney Slade
Assistant Editor: Mary St. Thomas
Editorial Assistant: Rachel Haskell
Senior Managing Editor: Karen Wernholm
Senior Production Project Manager: Kathleen A. Manley
Senior Author Support/Technology Specialist: Joe Vetere
Digital Assets Manager: Marianne Groth
Rights and Permissions Advisor: Michael Joyce
Image Manager: Rachel Youdelman
Media Producer: Lin Mahoney and Stephanie Green
Software Development: Kristina Evans and Mary Durnwald
Marketing Manager: Adam Goldstein
Marketing Assistant: Ashley Bryan
Design Manager: Andrea Nix
Cover Designer: Beth Paquin
Cover Art: High Pitch of Autumn by Gregory Packard Fine Art LLC, www.gregorypackard.com
Senior Manufacturing Buyer: Carol Melville
Senior Media Buyer: Ginny Michaud
Interior Design, Production Coordination, Composition, and Illustrations: Nesbitt
Graphics, Inc.
For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C-1, which is hereby made part of this copyright page.
Many of the designations used by manufacturers and sellers to distinguish their products are
claimed as trademarks. Where those designations appear in this book, and Addison-Wesley
was aware of a trademark claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Lial, Margaret L.
Algebra for college students/Margaret L. Lial, John Hornsby, Terry McGinnis.—7th ed.
p. cm.
ISBN-13: 978-0-321-71540-1
(student edition)
ISBN-10: 0-321-71540-3
(student edition)
1. Algebra—Textbooks. I. Hornsby, E. John. II. McGinnis, Terry. III. Title.
QA154.3.L53 2012
512.9—dc22
2010002284
NOTICE:
This work is
protected by U.S.
copyright laws and
is provided solely for
the use of college instructors in reviewing
course materials for
classroom use. Dissemination or sale of
this work, or any part
(including on the
World Wide Web),
will destroy the integrity of the work
and is not permitted.
The work and materials from it should
never be made available to students except by instructors
using the accompanying text in their
classes. All recipients of this work are
expected to abide by
these restrictions
and to honor the intended pedagogical
purposes and the
needs of other instructors who rely on
these materials.
Copyright © 2012, 2008, 2004, 2000 Pearson Education, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the publisher. Printed in the
United States of America. For information on obtaining permission for use of material in this
work, please submit a written request to Pearson Education, Inc., Rights and Contracts
Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to
617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—CRK—14 13 12 11 10
www.pearsonhighered.com
ISBN 13: 978-0-321-71540-1
ISBN 10: 0-321-71540-3
To my friends Brian and Denise Altobello
John
To Papa
T.
This page intentionally left blank
Contents
Preface
xiii
STUDY SKILLS
1
Using Your Math Textbook
xxii
Review of the Real Number System
1
1.1
1.2
1.3
1.4
Basic Concepts
2
Operations on Real Numbers
14
Exponents, Roots, and Order of Operations
Properties of Real Numbers
24
32
Chapter 1 Summary 39
Chapter 1 Review Exercises 42
Chapter 1 Test
STUDY SKILLS
2
44
Reading Your Math Textbook
46
Linear Equations, Inequalities, and Applications
47
2.1 Linear Equations in One Variable 48
STUDY SKILLS Tackling Your Homework 56
2.2 Formulas and Percent 56
2.3 Applications of Linear Equations 67
STUDY SKILLS Taking Lecture Notes 80
2.4 Further Applications of Linear Equations 81
SUMMARY EXERCISES on Solving Applied Problems 89
2.5 Linear Inequalities in One Variable 91
STUDY SKILLS Using Study Cards 102
2.6 Set Operations and Compound Inequalities 103
STUDY SKILLS Using Study Cards Revisited 111
2.7 Absolute Value Equations and Inequalities 112
SUMMARY EXERCISES on Solving Linear and Absolute Value Equations and
Inequalities 121
STUDY SKILLS Reviewing a Chapter 122
Chapter 2 Summary 123
Chapter 2 Review Exercises 127
Chapter 2 Test 131
Chapters 1–2 Cumulative Review Exercises 133
3
Graphs, Linear Equations, and Functions
135
3.1 The Rectangular Coordinate System 136
STUDY SKILLS Managing Your Time 147
3.2 The Slope of a Line 148
vii
viii
Contents
3.3 Linear Equations in Two Variables 161
SUMMARY EXERCISES on Slopes and Equations of Lines 174
3.4 Linear Inequalities in Two Variables 175
3.5 Introduction to Relations and Functions 181
3.6 Function Notation and Linear Functions 190
STUDY SKILLS Taking Math Tests 198
Chapter 3 Summary 199
Chapter 3 Review Exercises 202
Chapter 3 Test 205
Chapters 1–3 Cumulative Review Exercises 206
4
Systems of Linear Equations
209
4.1 Systems of Linear Equations in Two Variables 210
STUDY SKILLS Analyzing Your Test Results 225
4.2 Systems of Linear Equations in Three Variables 226
4.3 Applications of Systems of Linear Equations 233
4.4 Solving Systems of Linear Equations by Matrix Methods 247
Chapter 4 Summary 253
Chapter 4 Review Exercises 257
Chapter 4 Test 260
Chapters 1–4 Cumulative Review Exercises 261
5
Exponents, Polynomials, and Polynomial Functions
5.1
5.2
5.3
5.4
5.5
Integer Exponents and Scientific Notation
Adding and Subtracting Polynomials
264
278
Polynomial Functions, Graphs, and Composition
Multiplying Polynomials
Dividing Polynomials
284
293
302
Chapter 5 Summary 308
Chapter 5 Review Exercises 311
Chapter 5 Test
314
Chapters 1–5 Cumulative Review Exercises 316
6
Factoring
319
6.1
6.2
6.3
6.4
6.5
Greatest Common Factors and Factoring by Grouping
Factoring Trinomials
Special Factoring
326
333
A General Approach to Factoring
Solving Equations by Factoring
339
343
Chapter 6 Summary 354
Chapter 6 Review Exercises 356
Chapter 6 Test
358
Chapters 1–6 Cumulative Review Exercises 359
320
263
Contents
7
Rational Expressions and Functions
361
7.1 Rational Expressions and Functions; Multiplying and Dividing 362
7.2 Adding and Subtracting Rational Expressions 371
7.3 Complex Fractions 380
7.4 Equations with Rational Expressions and Graphs 386
SUMMARY EXERCISES on Rational Expressions and Equations 394
7.5 Applications of Rational Expressions 396
7.6 Variation 407
Chapter 7 Summary 416
Chapter 7 Review Exercises 420
Chapter 7 Test 423
Chapters 1–7 Cumulative Review Exercises 425
8
Roots, Radicals, and Root Functions
427
8.1 Radical Expressions and Graphs 428
8.2 Rational Exponents 435
8.3 Simplifying Radical Expressions 443
8.4 Adding and Subtracting Radical Expressions 453
8.5 Multiplying and Dividing Radical Expressions 458
SUMMARY EXERCISES on Operations with Radicals and Rational Exponents 466
8.6 Solving Equations with Radicals 468
8.7 Complex Numbers 474
STUDY SKILLS Preparing for Your Math Final Exam 482
Chapter 8 Summary 483
Chapter 8 Review Exercises 487
Chapter 8 Test 490
Chapters 1–8 Cumulative Review Exercises 492
9
Quadratic Equations and Inequalities
495
9.1 The Square Root Property and Completing the Square 496
9.2 The Quadratic Formula 505
9.3 Equations Quadratic in Form 512
SUMMARY EXERCISES on Solving Quadratic Equations 522
9.4 Formulas and Further Applications 523
9.5 Polynomial and Rational Inequalities 531
Chapter 9 Summary 537
Chapter 9 Review Exercises 540
Chapter 9 Test 543
Chapters 1–9 Cumulative Review Exercises 544
ix
x
Contents
10
Additional Graphs of Functions and Relations
10.1
10.2
10.3
10.4
10.5
Review of Operations and Composition
Graphs of Quadratic Functions
548
556
More About Parabolas and Their Applications
566
Symmetry; Increasing and Decreasing Functions
Piecewise Linear Functions
547
577
585
Chapter 10 Summary 594
Chapter 10 Review Exercises 597
Chapter 10 Test
600
Chapters 1–10 Cumulative Review Exercises 602
11
Inverse, Exponential, and Logarithmic Functions
11.1
11.2
11.3
11.4
11.5
11.6
Inverse Functions
606
Exponential Functions
614
Logarithmic Functions
622
Properties of Logarithms
629
Common and Natural Logarithms
638
Exponential and Logarithmic Equations; Further Applications
Chapter 11 Summary 657
Chapter 11 Review Exercises 660
Chapter 11 Test
664
Chapters 1–11 Cumulative Review Exercises 666
12
Polynomial and Rational Functions
669
12.1 Zeros of Polynomial Functions (I) 670
12.2 Zeros of Polynomial Functions (II) 676
12.3 Graphs and Applications of Polynomial Functions 685
SUMMARY EXERCISES on Polynomial Functions and Graphs 699
12.4 Graphs and Applications of Rational Functions 700
Chapter 12 Summary 714
Chapter 12 Review Exercises 717
Chapter 12 Test 720
Chapters 1–12 Cumulative Review Exercises 721
13
605
Conic Sections and Nonlinear Systems
725
13.1 The Circle and the Ellipse 726
13.2 The Hyperbola and Functions Defined by Radicals 734
13.3 Nonlinear Systems of Equations 741
647
Contents
13.4 Second-Degree Inequalities, Systems of Inequalities, and
Linear Programming 748
Chapter 13 Summary 757
Chapter 13 Review Exercises 760
Chapter 13 Test 763
Chapters 1–13 Cumulative Review Exercises 764
14
Further Topics in Algebra
14.1
14.2
14.3
14.4
14.5
14.6
14.7
767
Sequences and Series
768
Arithmetic Sequences
774
Geometric Sequences
781
The Binomial Theorem
791
Mathematical Induction
Counting Theory
796
801
Basics of Probability
809
Chapter 14 Summary 817
Chapter 14 Review Exercises 821
Chapter 14 Test
824
Chapters 1–14 Cumulative Review Exercises 825
Appendix A Properties of Matrices 827
Appendix B Matrix Inverses 837
Appendix C Determinants and Cramer’s Rule 847
Answers to Selected Exercises
Glossary
Credits
Index
G-1
C-1
I-1
A-1
xi
This page intentionally left blank
Preface
It is with pleasure that we offer the seventh edition of Algebra for College Students.
With each new edition, the text has been shaped and adapted to meet the changing
needs of both students and educators, and this edition faithfully continues that
process. As always, we have taken special care to respond to the specific suggestions
of users and reviewers through enhanced discussions, new and updated examples and
exercises, helpful features, updated figures and graphs, and an extensive package of
supplements and study aids. We believe the result is an easy-to-use, comprehensive
text that is the best edition yet.
Students who have never studied algebra—as well as those who require further review of basic algebraic concepts before taking additional courses in mathematics,
business, science, nursing, or other fields—will benefit from the text’s studentoriented approach. Of particular interest to students and instructors will be the
NEW Study Skills activities and Now Try Exercises.
This text is part of a series that also includes the following books:
N Beginning Algebra, Eleventh Edition, by Lial, Hornsby, and McGinnis
N Intermediate Algebra, Eleventh Edition, by Lial, Hornsby, and McGinnis
N Beginning and Intermediate Algebra, Fifth Edition, by Lial, Hornsby, and
McGinnis
NEW IN THIS EDITION
We are pleased to offer the following new student-oriented features and study aids:
Lial Video Library This collection of video resources helps students navigate the
road to success. It is available in MyMathLab and on Video Resources on DVD.
MyWorkBook This helpful guide provides extra practice exercises for every chapter of the text and includes the following resources for every section:
N Key vocabulary terms and vocabulary practice problems
N Guided Examples with step-by-step solutions and similar Practice Exercises,
keyed to the text by Learning Objective
N References to textbook Examples and Section Lecture Videos for additional help
N Additional Exercises with ample space for students to show their work, keyed to
the text by Learning Objective
Study Skills Poor study skills are a major reason why students do not succeed in
mathematics. In these short activities, we provide helpful information, tips, and
strategies on a variety of essential study skills, including Reading Your Math Textbook, Tackling Your Homework, Taking Math Tests, and Managing Your Time. While
most of the activities are concentrated in the early chapters of the text, each has been
designed independently to allow flexible use with individuals or small groups of students, or as a source of material for in-class discussions. (See pages 102 and 225.)
xiii
xiv
Preface
Now Try Exercises To actively engage students in the learning process, we now
include a parallel margin exercise juxtaposed with each numbered example. These allnew exercises enable students to immediately apply and reinforce the concepts and
skills presented in the corresponding examples. Answers are conveniently located on
the same page so students can quickly check their results. (See pages 3 and 92.)
Revised Exposition As each section of the text was being revised, we paid special
attention to the exposition, which has been tightened and polished. (See Section 5.2
Adding and Subtracting Polynomials, for example.) We believe this has improved
discussions and presentations of topics.
Specific Content Changes These include the following:
N We gave the exercise sets special attention. There are approximately 1250 new
and updated exercises, including problems that check conceptual understanding,
focus on skill development, and provide review. We also worked to improve the
even-odd pairing of exercises.
N Real-world data in over 170 applications in the examples and exercises have been
updated.
N There is an increased emphasis on the difference between expressions and equa-
tions, including a new example at the beginning of Section 2.1, plus corresponding exercises. Throughout the text, we have reformatted many example solutions
to use a “drop down” layout in order to further emphasize for students the difference between simplifying expressions and solving equations.
N We increased the emphasis on checking solutions and answers, as indicated by
the new CHECK tag and ✓ in the exposition and examples.
N Section 2.2 has been expanded to include a new example and exercises on solv-
ing a linear equation in two variables for y. A new objective, example, and exercises on percent increase and decrease are also provided.
N Section 3.5 Introduction to Functions from the previous edition has been ex-
panded and split into two sections.
N Key information about graphs is displayed prominently beside hand-drawn
graphs for the various types of functions. (See Sections 5.3, 7.4, 8.1, 10.2, 10.3,
10.5, 11.2, and 11.3.)
N An objective, example, and exercises on using factoring to solve formulas for
specified variables is included in Section 6.5.
N Presentations of the following topics have also been enhanced and expanded:
Solving three-part inequalities (Section 2.5)
Finding average rate of change (Section 3.2)
Writing equations of horizontal and vertical lines (Section 3.3)
Determining the number of solutions of a linear system (Section 4.1)
Solving systems of linear equations in three variables (Section 4.2)
Understanding the basic concepts and terminology of polynomials (Section 5.2)
Solving equations with rational expressions and graphing rational functions
(Section 7.4)
Solving quadratic equations by factoring and the square root property
(Section 9.1)
Preface
xv
Solving quadratic equations by substitution (Section 9.3)
Evaluating expressions involving the greatest integer (Section 10.5)
Graphing polynomial functions (Section 12.3)
Graphing hyperbolas (Section 13.2)
Solving linear programming problems (Section 13.4)
Evaluating factorials and binomial coefficients (Section 14.4)
HALLMARK FEATURES
We have included the following helpful features, each of which is designed to increase ease-of-use by students and/or instructors.
Annotated Instructor’s Edition For convenient reference, we include answers to
the exercises “on page” in the Annotated Instructor’s Edition, using an enhanced,
easy-to-read format. In addition, we have added approximately 15 new Teaching Tips
and over 40 new and updated Classroom Examples.
Relevant Chapter Openers In the new and updated chapter openers, we feature
real-world applications of mathematics that are relevant to students and tied to specific material within the chapters. Examples of topics include Americans’ spending
on pets, television ownership and viewing, and tourism. Each opener also includes a
section outline. (See pages 1, 47, and 263.)
Helpful Learning Objectives We begin each section with clearly stated, numbered
objectives, and the included material is directly keyed to these objectives so that students
and instructors know exactly what is covered in each section. (See pages 2 and 48.)
Popular Cautions and Notes One of the most popular features of previous
editions, we include information marked
CAUTION and NOTE to warn students
about common errors and emphasize important ideas throughout the exposition. The
updated text design makes them easy to spot. (See pages 53 and 140.)
Comprehensive Examples The new edition of this text features a multitude of
step-by-step, worked-out examples that include pedagogical color, helpful side comments, and special pointers. We give increased attention to checking example
solutions—more checks, designated using a special CHECK tag, are included than in
past editions. (See pages 51 and 270.)
More Pointers Well received by both students and instructors in the previous edition, we incorporate more pointers in examples and discussions throughout this
edition of the text. They provide students with important on-the-spot reminders and
warnings about common pitfalls. (See pages 96 and 396.)
Updated Figures, Photos, and Hand-Drawn Graphs Today’s students are
more visually oriented than ever. As a result, we have made a concerted effort to include appealing mathematical figures, diagrams, tables, and graphs, including a
“hand-drawn” style of graphs, whenever possible. (See pages 138 and 558.) Many of
the graphs also use a style similar to that seen by students in today’s print and electronic media. We have incorporated new photos to accompany applications in examples and exercises. (See pages 154 and 168.)
Relevant Real-Life Applications We include many new or updated applications
from fields such as business, pop culture, sports, technology, and the life sciences
that show the relevance of algebra to daily life. (See pages 76 and 244.)
xvi
Preface
Emphasis on Problem-Solving We introduce our six-step problem-solving
method in Chapter 2 and integrate it throughout the text. The six steps, Read, Assign
a Variable, Write an Equation, Solve, State the Answer, and Check, are emphasized
in boldface type and repeated in examples and exercises to reinforce the problemsolving process for students. (See pages 69 and 234.) We also provide students with
PROBLEM-SOLVING HINT boxes that feature helpful problem-solving tips and
strategies. (See pages 81 and 233.)
Connections We include these to give students another avenue for making connections to the real world, graphing technology, or other mathematical concepts, as well
as to provide historical background and thought-provoking questions for writing,
class discussion, or group work. (See pages 117 and 143.)
Ample and Varied Exercise Sets One of the most commonly mentioned strengths
of this text is its exercise sets. We include a wealth of exercises to provide students
with opportunities to practice, apply, connect, review, and extend the algebraic concepts and skills they are learning. We also incorporate numerous illustrations, tables,
graphs, and photos to help students visualize the problems they are solving. Problem
types include writing , graphing calculator , multiple-choice, true/false, matching,
and fill-in-the-blank problems, as well as the following:
N Concept Check exercises facilitate students’ mathematical thinking and concep-
tual understanding. (See pages 108 and 413.)
N WHAT WENT WRONG? exercises ask students to identify typical errors in solu-
tions and work the problems correctly. (See pages 274 and 502.)
N Brain Busters exercises challenge students to go beyond the section examples.
(See pages 145 and 300.)
N
RELATING CONCEPTS exercises help students tie together topics and develop
problem-solving skills as they compare and contrast ideas, identify and describe
patterns, and extend concepts to new situations. These exercises make great
collaborative activities for pairs or small groups of students. (See pages 173
and 301.)
N
TECHNOLOGY INSIGHTS exercises provide an opportunity for students to
interpret typical results seen on graphing calculator screens. Actual screens from
the TI-83/84 Plus graphing calculator are featured. (See pages 146 and 353.)
N
PREVIEW EXERCISES allow students to review previously-studied concepts
and preview skills needed for the upcoming section. These make good oral warmup exercises to open class discussions. (See pages 283 and 371.)
Special Summary Exercises We include a set of these popular in-chapter exercises in selected chapters. They provide students with the all-important mixed review
problems they need to master topics and often include summaries of solution methods and/or additional examples. (See pages 394 and 522.)
Extensive Review Opportunities We conclude each chapter with the following
review components:
N A Chapter Summary that features a helpful list of Key Terms, organized by
section, New Symbols, Test Your Word Power vocabulary quiz (with answers
immediately following), and a Quick Review of each section’s contents, complete with additional examples (See pages 483–486.)
Preface
xvii
N A comprehensive set of Chapter Review Exercises, keyed to individual sections
for easy student reference, as well as a set of Mixed Review Exercises that helps
students further synthesize concepts (See pages 487–490.)
N A Chapter Test that students can take under test conditions to see how well they
have mastered the chapter material (See pages 490–491.)
N A set of Cumulative Review Exercises (beginning in Chapter 2) that covers ma-
terial going back to Chapter 1 (See pages 492–493.)
Glossary For easy reference at the back of the book, we include a comprehensive
glossary featuring key terms and definitions from throughout the text. (See pages
G-1 to G-8.)
SUPPLEMENTS
For a comprehensive list of the supplements and study aids that accompany Algebra
for College Students, Seventh Edition, see pages xix–xxi.
ACKNOWLEDGMENTS
The comments, criticisms, and suggestions of users, nonusers, instructors, and students have positively shaped this textbook over the years, and we are most grateful for
the many responses we have received. Thanks to the following people for their review
work, feedback, assistance at various meetings, and additional media contributions:
Barbara Aaker, Community College of Denver
Viola Lee Bean, Boise State University
Kim Bennekin, Georgia Perimeter College
Dixie Blackinton, Weber State University
Tim Caldwell, Meridian Community College
Sally Casey, Shawnee Community College
Callie Daniels, St. Charles Community College
Cheryl Davids, Central Carolina Technical College
Chris Diorietes, Fayetteville Technical Community College
Sylvia Dreyfus, Meridian Community College
Lucy Edwards, Las Positas College
LaTonya Ellis, Bishop State Community College
Jacqui Fields, Wake Technical Community College
Beverly Hall, Fayetteville Technical Community College
Sandee House, Georgia Perimeter College
Lynette King, Gadsden State Community College
Linda Kodama, Windward Community College
Ted Koukounas, Suffolk Community College
Karen McKarnin, Allen County Community College
James Metz, Kapi´olani Community College
Jean Millen, Georgia Perimeter College
Molly Misko, Gadsden State Community College
William Remele, Brunswick Community College
Jane Roads, Moberly Area Community College
Melanie Smith, Bishop State Community College
xviii
Preface
Linda Smoke, Central Michigan University
Erik Stubsten, Chattanooga State Technical Community College
Tong Wagner, Greenville Technical College
Sessia Wyche, University of Texas at Brownsville
Special thanks are due the many instructors at Broward College who provided insightful comments.
Over the years, we have come to rely on an extensive team of experienced professionals. Our sincere thanks go to these dedicated individuals at Addison-Wesley, who
worked long and hard to make this revision a success: Chris Hoag, Maureen O’Connor, Michelle Renda, Adam Goldstein, Kari Heen, Courtney Slade, Kathy Manley,
Stephanie Green, Lin Mahoney, Rachel Haskell, and Mary St. Thomas.
We are especially grateful to Callie Daniels for her excellent work on the new Now
Try Exercises. Abby Tanenbaum did a terrific job helping us revise real-data applications. Kathy Diamond provided expert guidance through all phases of production and
rescued us from one snafu or another on multiple occasions. Marilyn Dwyer and
Nesbitt Graphics, Inc., provided some of the highest quality production work we have
experienced on the challenging format of these books.
Special thanks are due Jeff Cole, who continues to supply accurate, helpful solutions
manuals; David Atwood, who wrote the comprehensive Instructor’s Resource
Manual with Tests; Beverly Fusfield, who provided the new MyWorkBook; Beth
Anderson, who provided wonderful photo research; and Lucie Haskins, for yet another accurate, useful index. De Cook, Shannon d’Hemecourt, Paul Lorczak, and
Sarah Sponholz did a thorough, timely job accuracy checking manuscript and page
proofs. It has indeed been a pleasure to work with such an outstanding group of
professionals.
As an author team, we are committed to providing the best possible text and supplements package to help instructors teach and students succeed. As we continue to
work toward this goal, we would welcome any comments or suggestions you might
have via e-mail to math@pearson.com.
Margaret L. Lial
John Hornsby
Terry McGinnis
Preface
STUDENT SUPPLEMENTS
INSTRUCTOR SUPPLEMENTS
Student’s Solutions Manual
N By Jeffery A. Cole, Anoka-Ramsey Community College
N Provides detailed solutions to the odd-numbered,
Annotated Instructor’s Edition
N Provides “on-page” answers to all text exercises in
section-level exercises and to all Now Try Exercises,
Relating Concepts, Summary, Chapter Review,
Chapter Test, and Cumulative Review Exercises
xix
an easy-to-read margin format, along with Teaching
Tips and extensive Classroom Examples
N Includes icons to identify writing
and calculator
exercises. These are in the Student Edition also.
ISBNs: 0-321-71549-7, 978-0-321-71549-4
ISBNs: 0-321-71548-9, 978-0-321-71548-7
NEW Video Resources on DVD featuring the
Lial Video Library
N Provides a wealth of video resources to help stu-
Instructor’s Solutions Manual
N By Jeffery A. Cole, Anoka-Ramsey Community College
N Provides complete answers to all text exercises,
dents navigate the road to success
N Available in MyMathLab (with optional subtitles
in English)
N Includes the following resources:
Section Lecture Videos that offer a new navigation
menu for easy focus on key examples and exercises
needed for review in most sections (with optional
subtitles in Spanish)
Quick Review Lectures that provide a short summary
lecture of most key concepts from Quick Reviews
Chapter Test Prep Videos that include step-by-step
solutions to most Chapter Test exercises and give
guidance and support when needed most—the
night before an exam. Also available on YouTube
(searchable using author name and book title)
ISBNs: 0-321-71584-5, 978-0-321-71584-5
NEW MyWorkBook
N Provides Guided Examples and corresponding Now
Try Exercises for each text objective
N Refers students to correlated Examples, Lecture
including all Classroom Examples and Now Try
Exercises
ISBNs: 0-321-71543-8, 978-0-321-71543-2
Instructor’s Resource Manual with Tests
N By David Atwood, Rochester Community and Technical College
N Contains two diagnostic pretests, four free-response
and two multiple-choice test forms per chapter, and
two final exams
N Includes a mini-lecture for each section of the text
with objectives, key examples, and teaching tips
N Provides a correlation guide from the sixth to the
seventh edition
ISBNs: 0-321-71544-6, 978-0-321-71544-9
PowerPoint® Lecture Slides
N Present key concepts and definitions from the text
N Available for download at
www.pearsonhighered.com/irc
ISBNs: 0-321-71585-3, 978-0-321-71585-2
Videos, and Exercise Solution Clips
N Includes extra practice exercises for every section of
the text with ample space for students to show their
work
N Lists the learning objectives and key vocabulary
terms for every text section, along with vocabulary
practice problems
ISBNs: 0-321-71552-7, 978-0-321-71552-4
TestGen® (www.pearsonhighered.com/testgen)
N Enables instructors to build, edit, print, and administer tests using a computerized bank of questions
developed to cover all text objectives
N Allows instructors to create multiple but equivalent
versions of the same question or test with the click
of a button
N Allows instructors to modify test bank questions or
add new questions
N Available for download from Pearson Education’s
online catalog
ISBNs: 0-321-71545-4, 978-0-321-71545-6
xx
Preface
STUDENT SUPPLEMENTS
INSTRUCTOR SUPPLEMENTS
InterAct Math Tutorial Website
http://www.interactmath.com
N Provides practice and tutorial help online
N Provides algorithmically generated practice exercises
Pearson Math Adjunct Support Center
(http://www.pearsontutorservices.com/math-adjunct.
html)
N Staffed by qualified instructors with more than
50 years of combined experience at both the
community college and university levels
that correlate directly to the exercises in the textbook
N Allows students to retry an exercise with new values
each time for unlimited practice and mastery
N Includes an interactive guided solution for each exercise that gives helpful feedback when an incorrect
answer is entered
N Enables students to view the steps of a worked-out
sample problem similar to the one being worked on
Assistance is provided for faculty in the following
areas:
N
N
N
N
Suggested syllabus consultation
Tips on using materials packed with your book
Book-specific content assistance
Teaching suggestions, including advice on classroom
strategies
Available for Students and Instructors
MyMathLab® Online Course (Access code required.)
MyMathLab® is a text-specific, easily customizable online course that integrates
interactive multimedia instruction with textbook content. MyMathLab gives instructors the tools they need to deliver all or a portion of their course online, whether their
students are in a lab setting or working from home.
N Interactive homework exercises, correlated to the textbook at the objective
level, are algorithmically generated for unlimited practice and mastery. Most
exercises are free-response and provide guided solutions, sample problems, and
tutorial learning aids for extra help.
N Personalized homework assignments can be designed to meet the needs of
the class. MyMathLab tailors the assignment for each student based on their test
or quiz scores so that each student’s homework assignment contains only the
problems they still need to master.
N Personalized Study Plan, generated when students complete a test or quiz or
homework, indicates which topics have been mastered and links to tutorial exercises for topics students have not mastered. Instructors can customize the Study
Plan so that the topics available match their course content.
N Multimedia learning aids, such as video lectures and podcasts, animations,
and a complete multimedia textbook, help students independently improve their
understanding and performance. Instructors can assign these multimedia learning aids as homework to help their students grasp the concepts.
N Homework and Test Manager lets instructors assign homework, quizzes,
and tests that are automatically graded. They can select just the right mix of questions from the MyMathLab exercise bank, instructor-created custom exercises,
and/or TestGen® test items.
N Gradebook, designed specifically for mathematics and statistics, automatically
tracks students’ results, lets instructors stay on top of student performance, and
gives them control over how to calculate final grades. They can also add offline
(paper-and-pencil) grades to the gradebook.
Preface
xxi
N MathXL Exercise Builder allows instructors to create static and algorithmic
exercises for their online assignments. They can use the library of sample exercises as an easy starting point, or they can edit any course-related exercise.
N Pearson Tutor Center (www.pearsontutorservices.com) access is automati-
cally included with MyMathLab. The Tutor Center is staffed by qualified math
instructors who provide textbook-specific tutoring for students via toll-free
phone, fax, email, and interactive Web sessions.
Students do their assignments in the Flash®-based MathXL Player, which is compatible with almost any browser (Firefox®, SafariTM, or Internet Explorer®) on almost any
platform (Macintosh® or Windows®). MyMathLab is powered by CourseCompassTM,
Pearson Education’s online teaching and learning environment, and by MathXL®, our
online homework, tutorial, and assessment system. MyMathLab is available to qualified adopters. For more information, visit our website at www.mymathlab.com or
contact your Pearson representative.
MathXL® Online Course (Access code required.)
MathXL® is an online homework, tutorial, and assessment system that accompanies
Pearson’s textbooks in mathematics or statistics.
N Interactive homework exercises, correlated to the textbook at the objective
level, are algorithmically generated for unlimited practice and mastery. Most
exercises are free-response and provide guided solutions, sample problems, and
learning aids for extra help.
N Personalized homework assignments are designed by the instructor to meet
the needs of the class, and then personalized for each student based on their test
or quiz results. As a result, each student receives a homework assignment that
contains only the problems they still need to master.
N Personalized Study Plan, generated when students complete a test or quiz or
homework, indicates which topics have been mastered and links to tutorial exercises for topics students have not mastered. Instructors can customize the available topics in the study plan to match their course concepts.
N Multimedia learning aids, such as video lectures and animations, help stu-
dents independently improve their understanding and performance. These are
assignable as homework, to further encourage their use.
N Gradebook, designed specifically for mathematics and statistics, automatically
tracks students’ results, lets instructors stay on top of student performance, and
gives them control over how to calculate final grades.
N MathXL Exercise Builder allows instructors to create static and algorithmic
exercises for their online assignments. They can use the library of sample exercises as an easy starting point or the Exercise Builder to edit any of the courserelated exercises.
N Homework and Test Manager lets instructors create online homework,
quizzes, and tests that are automatically graded. They can select just the right mix
of questions from the MathXL exercise bank, instructor-created custom exercises, and/or TestGen test items.
The new, Flash®-based MathXL Player is compatible with almost any browser
(Firefox®, SafariTM, or Internet Explorer®) on almost any platform (Macintosh® or
Windows®). MathXL is available to qualified adopters. For more information, visit
our website at www.mathxl.com, or contact your Pearson representative.
xxii
Preface
SKILLS
STUDY
Using Your Math Textbook
Your textbook is a valuable resource. You will learn more if you fully make use of the
features it offers.
SECTIO N 2.4
General Features
N Table of Contents Find this at the front of the text. Mark
the chapters and sections you will cover, as noted on your
course syllabus.
N Answer Section Tab this section at the back of the book
so you can refer to it frequently when doing homework.
Answers to odd-numbered section exercises are provided.
Answers to ALL summary, chapter review, test, and cumulative review exercises are given.
N Glossary Find this feature after the answer section at the
2.4
OBJE CTIV ES
1
Solve problems
about different
denominations
of money.
2 Solve problems
about uniform
motion.
3 Solve problems
about angles.
NOW TRY
EXERC ISE 1
Steven Danielson has a
collection of 52 coins worth
$3.70. His collection contai
ns
only dimes and nickels.
How
many of each type of coin
does he have?
OBJE CTIV E 1
HINT
In problems involving mone
y, use the following basic
fact.
number of monetary
units of the same kind : denomination ⴝ total monetary
value
30 dimes have a monetary
value of 301$0.102 = $3.00
.
Fifteen 5-dollar bills have
a value of 151$52 = $75.
EXAM PLE 1
Solving a Money Deno
mina
tion Problem
For a bill totaling $5.65
, a cashier received 25 coins
consisting of nickels and
ters. How many of each
quardenomination of coin did
the cashier receive?
Step 1 Read the probl
em. The problem asks that
we find the number of nicke
the number of quarters the
ls and
cashier received.
Step 2 Assign a varia
ble. Then organize the inform
ation in a table.
Let
x = the number of nicke
ls.
Then 25 - x = the numb
er of quarters.
Nickels
Number of Coins
Step 4 Solve.
Value
0.05
0.05x
0.25
0.25125 - x2
5.65
Total
the last column of the table.
0.05x + 0.25125 - x2
= 5.65
0.05x + 0.25125 - x2
= 5.65
5x + 25125 - x2 = 565
Move decimal
5x + 625 - 25x = 565
points 2 places
to the right.
Denomination
x
25 - x
Step 3 Write an equat
ion from
helpful list of geometric formulas, along with review
information on triangles and angles. Use these for reference throughout the course.
- 20x = - 60
Multiply by 100.
Distributive property
Subtract 625. Combine
like terms.
Divide by - 20.
x = 3
Step 5 State the answ
er. The cashier has 3 nicke
ls and 25 - 3 = 22 quart
ers.
Step 6 Check. The cashie
r has 3 + 22 = 25 coins
, and the value of the coins
is
$0.05132 + $0.251222
= $5.65, as required.
Specific Features
NOW TRY
ION Be sure that your
answer is reasonable when
lems like Example 1. Becau
you are working probse you are dealing with
a number of coins, the corre
answer can be neither negat
ct
ive nor a fraction.
CAUT
NOW TRY ANSW ER
1. 22 dimes; 30 nickels
each section and again within the section as the corresponding material is presented. Once you finish a
section, ask yourself if you have accomplished them.
N Now Try Exercises These margin exercises allow you to immediately practice the
material covered in the examples and prepare you for the exercises. Check your
results using the answers at the bottom of the page.
N Pointers These small shaded balloons provide on-the-spot warnings and reminders,
point out key steps, and give other helpful tips.
N Cautions These provide warnings about common errors that students often make
or trouble spots to avoid.
N Notes These provide additional explanations or emphasize important ideas.
N Problem-Solving Hints These green boxes give helpful tips or strategies to use
Find an example of each of these features in your textbook.
81
Solve problems abou
t different denomina
tions of money.
PROB LEM- SOLV ING
Quarters
N List of Formulas Inside the back cover of the text is a
when you work applications.
Equations
Further Applications
of Linear Equations
back of the text. It provides an alphabetical list of the
key terms found in the text, with definitions and section
references.
N Objectives The objectives are listed at the beginning of
Further Applications
of Linear
CHAPTER
Review of the Real
Number System
1.1
Basic Concepts
1.2
Operations on Real
Numbers
1.3
Exponents, Roots,
and Order of
Operations
1.4
Properties of Real
Numbers
1
Americans love their pets. Over 71 million U.S. households owned pets in 2008.
Combined, these households spent more than $44 billion pampering their animal
friends. The fastest-growing segment of the pet industry is the high-end luxury area,
which includes everything from gourmet pet foods, designer toys, and specialty
furniture to groomers, dog walkers, boarding in posh pet hotels, and even pet
therapists. (Source: American Pet Products Manufacturers Association.)
In Exercise 101 of Section 1.3, we use an algebraic expression, one of the
topics of this chapter, to determine how much Americans have spent annually on
their pets in recent years.
1
2
CHAPTER 1
1.1
Review of the Real Number System
Basic Concepts
OBJECTIVES
1
Write sets using set
notation.
2
3
Use number lines.
Know the common
sets of numbers.
4
Find additive
inverses.
Use absolute value.
Use inequality
symbols.
Graph sets of real
numbers.
5
6
7
OBJECTIVE 1 Write sets using set notation. A set is a collection of objects
called the elements or members of the set. In algebra, the elements of a set are usually numbers. Set braces, { }, are used to enclose the elements.
For example, 2 is an element of the set 51, 2, 36. Since we can count the number
of elements in the set 51, 2, 36, it is a finite set.
In our study of algebra, we refer to certain sets of numbers by name. The set
N ⴝ 51, 2, 3, 4, 5, 6,
Á6
Natural (counting) numbers
is called the natural numbers, or the counting numbers. The three dots (ellipsis
points) show that the list continues in the same pattern indefinitely. We cannot list all
of the elements of the set of natural numbers, so it is an infinite set.
Including 0 with the set of natural numbers gives the set of whole numbers.
W ⴝ 50, 1, 2, 3, 4, 5, 6,
Á6
Whole numbers
The set containing no elements, such as the set of whole numbers less than 0, is called
the empty set, or null set, usually written 0 or { }.
CAUTION Do not write 506 for the empty set. 506 is a set with one element: 0.
Use the notation 0 or { } for the empty set.
To write the fact that 2 is an element of the set 51, 2, 36, we use the symbol 僆
(read “is an element of ”).
2 僆 51, 2, 36
The number 2 is also an element of the set of natural numbers N.
2僆N
To show that 0 is not an element of set N, we draw a slash through the symbol 僆.
0僆N
Two sets are equal if they contain exactly the same elements. For example,
51, 26 = 52, 16. (Order doesn’t matter.) However, 51, 26 Z 50, 1, 26 ( Z means “is
not equal to”), since one set contains the element 0 while the other does not.
In algebra, letters called variables are often used to represent numbers or to define sets of numbers. For example,
5x | x is a natural number between 3 and 156
(read “the set of all elements x such that x is a natural number between 3 and 15”)
defines the set
54, 5, 6, 7, Á , 146.
The notation 5x | x is a natural number between 3 and 156 is an example of setbuilder notation.
5x | x has property P6
⎧
⎪
⎨
⎪
⎩
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎧
⎪
⎨
⎪
⎩
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
the set of
all elements x
such that
x has a given property P
Basic Concepts
SECTION 1.1
NOW TRY
EXERCISE 1
List the elements in
5 p | p is a natural number less
than 66.
EXAMPLE 1
3
Listing the Elements in Sets
List the elements in each set.
(a) 5x | x is a natural number less than 46
The natural numbers less than 4 are 1, 2, and 3. This set is 51, 2, 36.
(b) 5x | x is one of the first five even natural numbers6 is 52, 4, 6, 8, 106.
(c) 5x | x is a natural number greater than or equal to 76
The set of natural numbers greater than or equal to 7 is an infinite set, written
with ellipsis points as
57, 8, 9, 10, Á 6.
NOW TRY
EXERCISE 2
Use set-builder notation to
describe the set.
59, 10, 11, 126
EXAMPLE 2
NOW TRY
Using Set-Builder Notation to Describe Sets
Use set-builder notation to describe each set.
(a) 51, 3, 5, 7, 96
There are often several ways to describe a set in set-builder notation. One way to
describe the given set is
5x | x is one of the first five odd natural numbers6.
(b) 55, 10, 15, Á 6
This set can be described as 5x | x is a multiple of 5 greater than 06.
NOW TRY
OBJECTIVE 2 Use number lines. A good way to get a picture of a set of numbers is to use a number line. See FIGURE 1 .
To draw a number line, choose
any point on the line and label it
0. Then choose any point to the
right of 0 and label it 1. Use the
distance between 0 and 1 as the
scale to locate, and then label,
other points.
The number 0 is neither positive nor negative.
Negative numbers
–5
–4
–3
–2
Positive numbers
–1
0
1
2
3
4
5
FIGURE 1
The set of numbers identified on the number line in FIGURE 1 , including positive
and negative numbers and 0, is part of the set of integers.
I ⴝ 5 Á , ⴚ3, ⴚ2, ⴚ1, 0, 1, 2, 3,
Á6
Integers
Each number on a number line is called the coordinate of the point that it labels,
while the point is the graph of the number. FIGURE 2 shows a number line with several
points graphed on it.
Graph of –1
–1
3
4
2
–3 –2 –1
NOW TRY ANSWERS
1. 51, 2, 3, 4, 56
2. 5x | x is a natural number
between 8 and 136
0
1
Coordinate
FIGURE 2
2
3
4
CHAPTER 1
Review of the Real Number System
The fractions - 12 and 34 , graphed on the number line in FIGURE 2 , are rational
numbers. A rational number can be expressed as the quotient of two integers, with
denominator not 0. The set of all rational numbers is written as follows.
e
p
` p and q are integers, q ⴝ 0 f
q
Rational numbers
The set of rational numbers includes the natural numbers, whole numbers, and integers, since these numbers can be written as fractions. For example,
14 =
14
,
1
-3
,
1
-3 =
0 =
and
0
.
1
A rational number written as a fraction, such as 18 or 23, can also be expressed as a
decimal by dividing the numerator by the denominator.
0.666 Á
32.000 Á
18
20
18
20
18
2
2
= 0.6
3
0.125
Terminating decimal
(rational number)
81.000
8
20
16
40
40
0
Remainder is 0.
1
= 0.125
8
Repeating decimal
(rational number)
Remainder is never 0.
A bar is written over
the repeating digit(s).
Thus, terminating decimals, such as 0.125 = 18, 0.8 = 45, and 2.75 = 11
4 , and repeating
2
3
decimals, such as 0.6 = 3 and 0.27 = 11, are rational numbers.
Decimal numbers that neither terminate nor repeat, which include many square
roots, are irrational numbers.
d
=C
d
is approximately
3.141592653....
FIGURE 3
22 = 1.414213562 Á
and
- 27 = - 2.6457513 Á
NOTE Some square roots, such as 216 = 4 and
9
225
Irrational numbers
= 35 , are rational.
Another irrational number is p, the ratio of the circumference of a circle to its diameter. See FIGURE 3 .
Some rational and irrational numbers are graphed on the number line in FIGURE 4 .
The rational numbers together with the irrational numbers make up the set of real
numbers. Every point on a number line corresponds to a real number, and every
real number corresponds to a point on the number line.
Real numbers
Irrational
numbers
–4
Rational
numbers
√2
–√7
–3
–2
–1
0
0.27
3
5
FIGURE 4
1
2
3
2.75
4
√16
SECTION 1.1
Basic Concepts
5
Know the common sets of numbers.
OBJECTIVE 3
Sets of Numbers
Natural numbers, or
counting numbers
Whole numbers
51, 2, 3, 4, 5, 6,
Á6
50, 1, 2, 3, 4, 5, 6,
Á6
5 Á , ⴚ3, ⴚ2, ⴚ1, 0, 1, 2, 3,
Integers
Rational numbers
p
Eq
Á6
p and q are integers, q ⴝ 0 F
Examples: 41 or 4, 1.3, - 92 or - 4 12 , 16
8 or 2, 9 or 3, 0.6
Irrational numbers
5x x is a real number that is not rational6
Examples: 3, - 2, p
5x x is a rational number or an irrational number6*
Real numbers
FIGURE 5 shows the set of real numbers. Every real number is either rational or irrational. Notice that the integers are elements of the set of rational numbers and that
the whole numbers and natural numbers are elements of the set of integers.
Real numbers
Rational numbers
4
–1
9
4
–0.125 1.5
11
7
0.18
Irrational numbers
2
–3 5
4
–
8
15
23
Integers
..., –3, –2, –1
π
π
4
Whole
numbers
0
Natural
numbers
1, 2, 3, ...
FIGURE 5
NOW TRY
EXERCISE 3
List the numbers in the
following set that are elements of each set.
E - 2.4, - 1, - 12 , 0, 0.3,
5, p, 5 F
(a) Whole numbers
(b) Rational numbers
NOW TRY ANSWERS
3. (a) 50, 56
(b) E - 2.4, - 1, - 12 , 0, 0.3, 5 F
EXAMPLE 3
Identifying Examples of Number Sets
List the numbers in the following set that are elements of each set.
e - 8, - 5, -
9
1
, 0, 0.5, , 1.12, 3, 2, p f
64
3
(a) Integers
- 8, 0, and 2
(b) Rational numbers
9
- 8, - 64
, 0, 0.5, 13 , 1.12, and 2
(c) Irrational numbers
- 5, 3, and p
(d) Real numbers
All are real numbers.
NOW TRY
*An example of a number that is not real is - 1. This number, part of the complex number system, is
discussed in Chapter 8.
6
Review of the Real Number System
CHAPTER 1
NOW TRY
EXERCISE 4
EXAMPLE 4
Decide whether each statement is true or false. If it is
false, tell why.
(a) All integers are irrational
numbers.
(b) Every whole number is an
integer.
Determining Relationships Between Sets of Numbers
Decide whether each statement is true or false.
(a) All irrational numbers are real numbers.
This is true. As shown in FIGURE 5 , the set of real numbers includes all irrational
numbers.
(b) Every rational number is an integer.
This statement is false. Although some rational numbers are integers, other
rational numbers, such as 23 and - 14 , are not.
NOW TRY
Find additive inverses. Look at
For each positive number, there is a negative
number on the opposite side of 0 that lies the same
distance from 0. These pairs of numbers are called
additive inverses, opposites, or negatives of each
other. For example, 3 and - 3 are additive inverses.
OBJECTIVE 4
FIGURE 6 .
–3 –2 –1
0
1
2
3
Additive inverses (opposites)
FIGURE 6
Additive Inverse
For any real number a, the number - a is the additive inverse of a.
We change the sign of a number to find its additive inverse. As we shall see later,
the sum of a number and its additive inverse is always 0.
Uses of the Symbol ⴚ
The symbol “ - ” is used to indicate any of the following:
1. a negative number, such as - 9 or - 15;
2. the additive inverse of a number, as in “ - 4 is the additive inverse of 4”;
3. subtraction, as in 12 - 3.
In the expression - 1- 52, the symbol “ - ” is being used in two ways. The first indicates the additive inverse (or opposite) of - 5, and the second indicates a negative
number, - 5. Since the additive inverse of - 5 is 5, it follows that
- 1- 52 = 5.
Number
Additive Inverse
6
-6
-4
4
2
3
- 23
- 8.7
8.7
0
0
The number 0 is its own additive
inverse.
NOW TRY ANSWERS
4. (a) false; All integers are
rational numbers.
(b) true
ⴚ1ⴚa2
For any real number a,
ⴚ1ⴚa2 ⴝ a.
Numbers written with positive or negative signs, such as +4, +8, - 9, and - 5,
are called signed numbers. A positive number can be called a signed number even
though the positive sign is usually left off. The table in the margin shows the additive
inverses of several signed numbers.
Use absolute value. Geometrically, the absolute value of a
number a, written | a |, is the distance on the number line from 0 to a. For example, the
absolute value of 5 is the same as the absolute value of - 5 because each number lies
five units from 0. See FIGURE 7 on the next page.
OBJECTIVE 5
SECTION 1.1
Distance is 5,
so ⏐–5⏐ = 5.
Basic Concepts
7
Distance is 5,
so ⏐5⏐ = 5.
–5
0
5
FIGURE 7
CAUTION Because absolute value represents distance, and distance is never
negative, the absolute value of a number is always positive or 0.
The formal definition of absolute value follows.
Absolute Value
For any real number a,
a ⴝ e
a if a is positive or 0
ⴚa if a is negative.
The second part of this definition, | a | = - a if a is negative, requires careful thought.
If a is a negative number, then - a, the additive inverse or opposite of a, is a positive
number. Thus, | a | is positive. For example, if a = - 3, then
| a | = | - 3 | = - 1- 32 = 3.
NOW TRY
EXERCISE 5
Simplify by finding each
absolute value.
(a) | - 7 |
(b) - | - 15 |
(c) | 4 | - | - 4 |
EXAMPLE 5
| a | = - a if a is negative.
Finding Absolute Value
Simplify by finding each absolute value.
(a) | 13 | = 13
(b) | - 2 | = - 1- 22 = 2
(c) | 0 | = 0
(d) | - 0.75 | = 0.75
(e) - | 8 | = - 182 = - 8
(f ) - | - 8 | = - 182 = - 8
Evaluate the absolute value.
Then find the additive inverse.
Work as in part (e); | - 8 | = 8.
(g) | - 2 | + | 5 | = 2 + 5 = 7
Evaluate each absolute value, and then add.
(h) - | 5 - 2 | = - | 3 | = - 3
Subtract inside the bars first.
EXAMPLE 6
NOW TRY
Comparing Rates of Change in Industries
The projected total rates of change in employment (in percent) in some of the fastestgrowing and in some of the most rapidly declining occupations from 2006 through
2016 are shown in the table.
Occupation (2006–2016)
Customer service representatives
NOW TRY ANSWERS
5. (a) 7
(b) - 15 (c) 0
Total Rate of Change
(in percent)
24.8
Home health aides
48.7
Security guards
16.9
Word processors and typists
- 11.6
File clerks
- 41.3
Sewing machine operators
- 27.2
Source: Bureau of Labor Statistics.
8
CHAPTER 1
Review of the Real Number System
NOW TRY
EXERCISE 6
Refer to the table in Example 6
on the preceding page. Of the
security guards, file clerks, and
customer service representatives, which occupation is expected to see the least change
(without regard to sign)?
What occupation in the table on the preceding page is expected to see the greatest
change? The least change?
We want the greatest change, without regard to whether the change is an increase
or a decrease. Look for the number in the table with the greatest absolute value. That
number is for home health aides, since | 48.7 | = 48.7. Similarly, the least change is
for word processors and typists: | - 11.6 | = 11.6.
NOW TRY
Use inequality symbols. The statement
OBJECTIVE 6
4 + 2 = 6
is an equation—a statement that two quantities are equal. The statement
4 Z 6
(read “4 is not equal to 6”)
is an inequality—a statement that two quantities are not equal.
If two numbers are not equal, one must be less than the other. When reading from
left to right, the symbol 6 means “is less than.”
8 6 9,
- 6 6 15,
0 6
- 6 6 - 1, and
4
3
All are true.
Reading from left to right, the symbol 7 means “is greater than.”
12 7 5,
9 7 - 2,
- 4 7 - 6,
6
7 0
5
and
All are true.
In each case, the symbol “points” toward the lesser number.
The number line in FIGURE 8 shows the graphs of the numbers 4 and 9. We know
that 4 6 9. On the graph, 4 is to the left of 9. The lesser of two numbers is always to
the left of the other on a number line.
4