net ELEMENTARY AND INTERMEDIATE FOURTH EDITION ALGEBRA Alan S. David Gustafson Citrus College Rock Valley College Harry Shrunk © 1972 Christo Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States www.net Elementary and Intermediate Algebra, © 2009, 2005 Brooks/Cole, Cengage Learning Fourth Edition ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be Alan S. David Gustafson reproduced, transmitted, stored, or used in any form or by any means, graphic, electronic, or Executive Editor: mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Charlie Van Wagner Web distribution, information networks, or information storage and retrieval systems, except as Development Editor: permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior Danielle Derbenti written permission of the publisher.
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Permissions Editor: Purchase any of our products at your local college store or at our preferred online store Bob Kauser www. Production Service: Graphic World Inc. Text Designer: Terri Wright Photo Researcher: Terri Wright Illustrator: Lori Heckelman Cover Designer: Terri Wright Cover Image: Harry Shrunk © 1972 Christo Cover Printer: Transcontinental Printing / Interglobe Compositor: Graphic World Inc. Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08 www.net In memory of my mother, Jeanene, and in honor of my dad, Bill.
—AST In memory of my teacher and mentor, Professor John Finch.net This page intentionally left blank www.net CONTENTS 1 AN INTRODUCTION TO ALGEBRA 1 1.1 Introducing the Language of Algebra 2 1.3 The Real Numbers 25 1.4 Adding Real Numbers; Properties of Addition 35 1.5 Subtracting Real Numbers 44 1.6 Multiplying and Dividing Real Numbers; Multiplication and Division Properties 51 1.7 Exponents and Order of Operations 62 1.9 Simplifying Algebraic Expressions Using Properties of Real Numbers 85 Chapter Summary and Review 96 Chapter Test 105 Group Project 106 2 EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING 107 2.1 Solving Equations Using Properties of Equality 108 2.2 More about Solving Equations 119 2.3 Applications of Percent 129 2.6 More about Problem Solving 162 2.7 Solving Inequalities 173 Chapter Summary and Review 187 Chapter Test 194 Group Project 196 3 GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES; FUNCTIONS 197 3.1 Graphing Using the Rectangular Coordinate System 198 3.2 Graphing Linear Equations 208 3.4 Slope and Rate of Change 232 3.5 Slope–Intercept Form 247 3.6 Point–Slope Form 258 3.7 Graphing Linear Inequalities 268 3.8 An Introduction to Functions 279 Chapter Summary and Review 293 Chapter Test 302 Group Project 304 Cumulative Review 305 v www.net vi CONTENTS 4 SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 307 4.1 Solving Systems of Equations by Graphing 308 4.2 Solving Systems of Equations by Substitution 319 4.3 Solving Systems of Equations by Elimination (Addition) 330 4.4 Problem Solving Using Systems of Equations 341 4.5 Solving Systems of Linear Inequalities 356 Chapter Summary and Review 366 Chapter Test 372 Group Project 373 5 EXPONENTS AND POLYNOMIALS 375 5.1 Rules for Exponents 376 5.2 Zero and Negative Exponents 389 5.5 Adding and Subtracting Polynomials 416 5.8 Dividing Polynomials 446 Chapter Summary and Review 456 Chapter Test 464 Group Project 465 Cumulative Review 466 6 FACTORING AND QUADRATIC EQUATIONS 469 6.1 The Greatest Common Factor; Factoring by Grouping 470 6.2 Factoring Trinomials of the Form x2 bx c 481 6.3 Factoring Trinomials of the Form ax 2 bx c 493 6.4 Factoring Perfect-Square Trinomials and the Differences of Two Squares 504 6.5 Factoring the Sum and Difference of Two Cubes 512 6.7 Solving Quadratic Equations by Factoring 522 6.8 Applications of Quadratic Equations 531 Chapter Summary and Review 540 Chapter Test 547 Group Project 549 7 RATIONAL EXPRESSIONS AND EQUATIONS 551 7.1 Simplifying Rational Expressions 552 7.2 Multiplying and Dividing Rational Expressions 562 7.3 Adding and Subtracting with Like Denominators; Least Common Denominators 572 7.4 Adding and Subtracting with Unlike Denominators 582 7.5 Simplifying Complex Fractions 591 7.6 Solving Rational Equations 600 7.7 Problem Solving Using Rational Equations 609 7.8 Proportions and Similar Triangles 620 Chapter Summary and Review 632 Chapter Test 640 Group Project 642 Cumulative Review 642 www.net CONTENTS vii 8 TRANSITION TO INTERMEDIATE ALGEBRA 645 8.1 Review of Solving Linear Equations, Formulas, and Linear Inequalities 646 8.2 Solving Compound Inequalities 660 8.3 Solving Absolute Value Equations and Inequalities 672 8.4 Review of Factoring Methods: GCF, Grouping, Trinomials 685 8.5 Review of Factoring Methods: The Difference of Two Squares; the Sum and Difference of Two Cubes 697 8.6 Review of Rational Expressions and Rational Equations 705 8.7 Review of Linear Equations in Two Variables 720 8.9 Variation 756 Chapter Summary and Review 765 Chapter Test 781 Group Project 783 Cumulative Review 784 9 RADICAL EXPRESSIONS AND EQUATIONS 787 9.1 Radical Expressions and Radical Functions 788 9.3 Simplifying and Combining Radical Expressions 817 9.4 Multiplying and Dividing Radical Expressions 829 9.5 Solving Radical Equations 842 9.6 Geometric Applications of Radicals 854 9.7 Complex Numbers 867 Chapter Summary and Review 880 Chapter Test 889 Group Project 891 10 QUADRATIC EQUATIONS, FUNCTIONS, AND INEQUALITIES 893 10.1 The Square Root Property and Completing the Square 894 10.2 The Quadratic Formula 907 10.3 The Discriminant and Equations That Can Be Written in Quadratic Form 919 10.4 Quadratic Functions and Their Graphs 929 10.5 Quadratic and Other Nonlinear Inequalities 945 Chapter Summary and Review 957 Chapter Test 964 Group Project 966 Cumulative Review 967 11 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 971 11.1 Algebra and Composition of Functions 972 11.4 Base-e Exponential Functions 1012 11.6 Base-e Logarithmic Functions 1035 11.7 Properties of Logarithms 1043 11.8 Exponential and Logarithmic Equations 1055 Chapter Summary and Review 1068 Chapter Test 1079 Group Project 1080 www.net viii CONTENTS 12 MORE ON SYSTEMS OF EQUATIONS 1083 12.1 Solving Systems of Equations in Two Variables 1084 12.2 Solving Systems of Equations in Three Variables 1103 12.3 Problem Solving Using Systems of Thee Equations 1114 12.4 Solving Systems of Equations Using Matrices 1122 12.5 Solving Systems of Equations Using Determinants 1134 Chapter Summary and Review 1145 Chapter Test 1154 Group Project 1155 Cumulative Review 1156 13 CONIC SECTIONS; MORE GRAPHING 1161 13.1 The Circle and the Parabola 1162 13.4 Solving Nonlinear Systems of Equations 1197 Chapter Summary and Review 1205 Chapter Test 1210 Group Project 1212 14 MISCELLANEOUS TOPICS 1213 14.1 The Binomial Theorem 1214 14.2 Arithmetic Sequences and Series 1224 14.3 Geometric Sequences and Series 1235 Chapter Summary and Review 1248 Chapter Test 1252 Group Project 1253 Cumulative Review 1254 APPENDIXES APPENDIX 1: Roots and Powers A-1 APPENDIX 2: Synthetic Division A-2 APPENDIX 3: Answers to Selected Exercises A-9 INDEX I-1 www.net PREFACE Elementary and Intermediate Algebra, Fourth Edition, is more than a simple upgrade of the third edition. Substantial changes have been made to the example structure, the Study Sets, and the pedagogy. Throughout the process, the objective has been to ease teaching challenges and meet students’ educational needs. Algebra, for many of today’s developmental math students, is like a foreign language.
They have difficulty translating the words, their meanings, and how they apply to problem solving. With these needs in mind (and as educational research suggests), the fundamental goal is to have students read, write, think, and speak using the language of algebra. Instruc- tional approaches that include vocabulary, practice, and well-defined pedagogy, along with an emphasis on reasoning, modeling, communication, and technology skills have been blended to address this need. The most common student question as they watch their instructors solve problems and as they read the textbook is.
Why? The new fourth edition addresses this question in a unique way. Experience teaches us that it’s not enough to know how a problem is solved. Students gain a deeper understanding of algebraic concepts if they know why a particular approach is taken. This instructional truth was the motivation for adding a Strategy and Why explanation to the solution of each worked example.
The fourth edition now provides, on a consistent basis, a concise answer to that all-important question: Why? This is just one of several changes in this revision, and we trust that all of them will make the course a better experience for both instructor and student. NEW TO THIS EDITION • New Example Structure • New Chapter Opening Applications • New Study Skills Workshops • New Chapter Objectives • New Guided Practice and Try It Yourself sections in the Study Sets • New End-of-Chapter Organization ix www.net x PREFACE Chapter Openers Answering The Systems of Linear Equations Question: When Will I Use This? and Inequalities Have you heard this question before? Instructors are 4.1 Solving Systems of Equations by Graphing asked this question time and again by students.2 Solving Systems of Equations by response, we have written chapter openers called From Substitution 4.3 Solving Systems of Equations by Campus to Careers. This feature highlights vocations Elimination (Addition) 4.4 Problem Solving Using Systems of that require various algebraic skills. Designed to inspire Equations 4.5 Solving Systems of Linear career exploration, each includes job outlook, educa- Inequalities CHAPTER SUMMARY AND REVIEW tional requirements, and annual earnings information.
CHAPTER TEST Group Project Careers presented in the openers are tied to an exercise found later in the Study Sets. Peter Steiner/Alamy Examples That Tell Students Not Just How, But WHY from Campus to Careers Why? That question is often asked by students as Portrait Photographer Portrait photographers take pictures of individuals or groups of people and often they watch their instructor solve problems in class work in their own studios. Some specialize in weddings, religious ceremonies, or school photographs, and many work on location. Their job responsibilities require and as they are working on problems at home.
a variety of mathematical skills such as scheduling appointments, keeping 4a ⫹ 7b ⫽billing ⫺8 customers, EXAMPLE 4 Solve the system: e financial records, pricing photographs, purchasing supplies, It’s not enough to know how a problem is solved. and operating digital equipment. 5a ⫹ 6b ⫽ 1 Photographers often make packets of pictures available to their customers. Students gain a deeper understanding of the alge- StrategyIn Problem We will use 29 of the Set Study elimination method 4.4, we will find to solve the costs of two this sizes system.
of photo- graphs that are part of a wedding picture packet. braic concepts if they know why a particular Why Since none of the variables has coefficient 1 or ⫺1, it would be difficult to solve this system using substitution. approach was taken. This instructional truth was Solution the motivation for adding a Strategy and Why Step 1: Both equations are written in standard Ax ⫹ By ⫽ C form.
307 explanation to each worked example. Step 2: In this example, we must write both equations in equivalent forms to obtain like terms that are opposites. To eliminate a, we can multiply the first equation by 5 to create the term 20a, and we can multiply the second equation by ⫺4 to create the term ⫺20a.
