Need help with algebra

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undefined algebraforcollegestudent week 8 discussion.docx special product formulas are very important especially as you continue to work with polynomials. Choose an exercise from the following questions undefined where you multiply the polynomials using a special product rule learned on pgs. 295-297. Show how to multiply using the special product rule and then also show how you would multiply the same polynomials using the distribution, vertical, or box method from pgs. 293-294. Note the exercise number in your post. Explain which method you prefer and why?


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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
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