Mastery Learning Proven Results ALEKS is a unique, online program that uses artificial intelligence and adaptive questioning proven to raise student proficiency and success rates in math ALEKS Delivers a Unique Math Experience: Research-Based, Artificial Intelligence precisely measures each student’s knowledge Individualized Learning presents the exact topics each student is most ready to learn Adaptive, Open-Response Environment includes comprehensive tutorials and resources Detailed, Automated Reports track student and class progress toward course mastery Course Management Tools include textbook integration, custom features, and more The ALEKS Pie summarizes a student’s current knowledge and then delivers an individualized learning path with the exact topics the student is most ready to learn Dark portion represents what the student knows Light portion represents what the student still has to learn Ready to Learn topics appear in pop-up boxes when the student scrolls over a pie slice With ALEKS 360, the multimedia eBook is connected to every problem so students can quickly review the exact section they are working on “My experience with ALEKS has been effective, efficient, and eloquent Our students’ pass rates improved from 49 percent to 82 percent with ALEKS We also saw student retention rates increase by 12% in the next course Students feel empowered as they guide their own learning through ALEKS.” —Professor Eden Donahou, Seminole State College of Florida ALEKS is a registered trademark of ALEKS Corporation To see ALEKS in action, please visit: www.SuccessInMath.com Visualizing Math Concepts Dynamic Math Animations The Miller/O’Neill/Hyde author team has developed a series of Flash animations to illustrate difficult concepts where static images and text fall short The animations leverage the use of on-screen movement and morphing shapes to enhance conceptual learning Writing a Linear Model Using Observed Data Points Example Writing a Linear Model from Observed Data Points The monthly sales of hybrid cars sold in the United States are given for a recent year The sales for the first months of the year are shown in Figure 3-35 The value x ϭ represents January, x ϭ represents February, and so on Number of Vehicles Sold y Monthly Hybrid Vehicle Sales in the United States 30,000 25,000 (5, 23400) 20,000 15,000 (0, 14400) 10,000 5,000 0 Month (x ϭ represents January) x Figure 3-35 Through their classroom experience, the authors recognize that such media assets are great teaching tools for the classroom and excellent for online learning The Miller/O’Neill/ Hyde animations are interactive and quite diverse in their use Some provide a virtual laboratory for which an application is simulated and where students can collect data points for analysis and modeling Others provide interactive question-and-answer sessions to test conceptual learning For word problem applications, the animations ask students to estimate answers and practice “number sense.” Algebra Intermediate THIRD EDITION N Julie Miller Daytona State College Molly O’Neill Daytona State College Nancy Hyde Broward College— Professor Emeritus INTERMEDIATE ALGEBRA, THIRD EDITION Published by McGraw-Hill Education, Penn Plaza, New York, NY 10121 Copyright © 2015 by McGrawHill Education All rights reserved Printed in the United States of America Previous editions © 2013, 2010, and 2007 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOW/DOW ISBN 978–0–07–338442–9 MHID 0–07–338442–9 ISBN 978–0–07–734297–5 (Annotated Instructor’s Edition) MHID 0–07–734297–6 Senior Vice President, Products & Markets: Kurt L Strand Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Production & Technology Services: Kimberly Meriwether David Managing Director: Ryan Blankenship Brand Manager: Mary Ellen Rahn Director of Marketing: Alex Gay Director of Development: Rose Koos Development Editor: Emily Williams Director of Digital Content: Nicole Lloyd Director, Content Production: Terri Schiesl Content Project Manager: Peggy Selle Buyer: Nicole Baumgartner Senior Designer: Laurie B Janssen Cover Illustration: Imagineering Media Services, Inc Lead Content Licensing Specialist: Carrie K Burger Compositor: Aptara ®, Inc Typeface: 10/12 Times Ten Roman Printer: R R Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Miller, Julie, 1962Intermediate algebra / Julie Miller, Molly O’Neill, Nancy Hyde.–Third edition pages cm Includes index ISBN 978–0–07–338442–9—ISBN 0–07–338442–9—ISBN 978–0–07–734297–5— ISBN 0–07–734297–6 (hard copy : alk paper) Algebra–Textbooks I O’Neill, Molly, 1953- II Hyde, Nancy III Title QA154.3.M554 2014b 512—dc23 2013026119 The Internet addresses listed in the text were accurate at the time of publication The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites www.mhhe.com About the Authors Julie Miller is from Daytona State College, where she has taught developmental and upper-level mathematics courses for 20 years Prior to her work at Daytona State College, she worked as a software engineer for General Electric in the area of flight and radar simulation Julie earned a bachelor of science in applied mathematics from Union College in Schenectady, New York, and a master of science in mathematics from the University of Florida In addition to her textbooks in developmental mathematics, Julie has authored a college algebra textbook and several course supplements for college algebra, trigonometry, and precalculus “My father is a medical researcher, and I got hooked on math and science when I was young and would visit his laboratory I can remember using graph paper to plot data points for his experiments and doing simple calculations He would then tell me what the peaks and features in the graph meant in the context of his experiment I think that applications and hands-on experience made math come alive for me and I’d like to see math come alive for my students.” —Julie Miller Molly O’Neill is also from Daytona State College, where she has taught for 22 years in the School of Mathematics She has taught a variety of courses from developmental mathematics to calculus Before she came to Florida, Molly taught as an adjunct instructor at the University of Michigan–Dearborn, Eastern Michigan University, Wayne State University, and Oakland Community College Molly earned a bachelor of science in mathematics and a master of arts and teaching from Western Michigan University in Kalamazoo, Michigan Besides this textbook, she has authored several course supplements for college algebra, trigonometry, and precalculus and has reviewed texts for developmental mathematics “I differ from many of my colleagues in that math was not always easy for me But in seventh grade I had a teacher who taught me that if I follow the rules of mathematics, even I could solve math problems Once I understood this, I enjoyed math to the point of choosing it for my career I now have the greatest job because I get to math every day and I have the opportunity to influence my students just as I was influenced Authoring these texts has given me another avenue to reach even more students.” —Molly O’Neill Nancy Hyde served as a full-time faculty member of the Mathematics Department at Broward College for 24 years During this time she taught the full spectrum of courses from developmental math through differential equations She received a bachelor of science degree in math education from Florida State University and a master’s degree in math education from Florida Atlantic University She has conducted workshops and seminars for both students and teachers on the use of technology in the classroom In addition to this textbook, she has authored a graphing calculator supplement for College Algebra “I grew up in Brevard County, Florida, where my father worked at Cape Canaveral I was always excited by mathematics and physics in relation to the space program As I studied higher levels of mathematics I became more intrigued by its abstract nature and infinite possibilities It is enjoyable and rewarding to convey this perspective to students while helping them to understand mathematics.” —Nancy Hyde Dedications To my Mom To my granddaughter, Kira To: Bella and Rosie —Nancy Hyde —Molly O’Neill —Julie Miller v Contents Chapter R Review of Basic Algebraic Concepts R.1 Study Skills Group Activity: Becoming a Successful Student R.2 Sets of Numbers and Interval Notation R.3 Operations on Real Numbers 16 R.4 Simplifying Algebraic Expressions 30 Chapter R Summary 37 Chapter R Review Exercises 39 Chapter R Test 41 Chapter Linear Equations and Inequalities in One Variable 43 1.1 Linear Equations in One Variable 44 Problem Recognition Exercises: Equations Versus Expressions 54 1.2 Applications of Linear Equations in One Variable 1.3 Applications to Geometry and Literal Equations 55 66 1.4 Linear Inequalities in One Variable 74 1.5 Compound Inequalities 82 1.6 Absolute Value Equations 93 1.7 Absolute Value Inequalities 99 Problem Recognition Exercises: Identifying Equations and Inequalities 109 Group Activity: Understanding the Symbolism of Mathematics Chapter Summary 110 111 Chapter Review Exercises 118 Chapter Test 121 Chapter Linear Equations in Two Variables 123 2.1 Linear Equations in Two Variables 124 2.2 Slope of a Line and Rate of Change 141 2.3 Equations of a Line 152 Problem Recognition Exercises: Identifying Characteristics of Lines 165 2.4 Applications of Linear Equations and Modeling 165 Group Activity: Using Linear Equations to Construct Images Chapter Summary 178 Chapter Review Exercises 182 Chapter Test 185 Chapters 1–2 Cumulative Review Exercises vi 187 176 Chapter Relations and Functions 189 3.1 Relations and Applications 190 3.2 Introduction to Functions 198 3.3 Graphs of Basic Functions 210 Problem Recognition Exercises: Characteristics of Relations 222 3.4 Algebra of Functions, Composition, and Applications 223 Group Activity: Deciphering a Coded Message 229 Chapter Summary 230 Chapter Review Exercises 233 Chapter Test 236 Chapters 1–3 Cumulative Review Exercises 237 Chapter Systems of Linear Equations 239 4.1 Solving Systems of Linear Equations by the Graphing Method 240 4.2 Solving Systems of Equations by Using the Substitution Method 249 4.3 Solving Systems of Equations by Using the Addition Method 256 Problem Recognition Exercises: Solving Systems of Linear Equations 263 4.4 Applications of Systems of Linear Equations in Two Variables 264 4.5 Linear Inequalities and Systems of Linear Inequalities in Two Variables 273 4.6 Systems of Linear Equations in Three Variables and Applications 286 4.7 Solving Systems of Linear Equations by Using Matrices 295 Group Activity: Creating a Quadratic Model of the Form y ؍at2 ؉ bt ؉ c 304 Chapter Summary 305 Chapter Review Exercises 311 Chapter Test 315 Chapters 1–4 Cumulative Review Exercises 317 Chapter Polynomials 319 5.1 5.2 5.3 5.4 Properties of Integer Exponents and Scientific Notation 320 Addition and Subtraction of Polynomials and Polynomial Functions 329 Multiplication of Polynomials 339 Division of Polynomials 349 Problem Recognition Exercises: Operations on Polynomials 358 5.5 Greatest Common Factor and Factoring by Grouping 359 5.6 Factoring Trinomials and Perfect Square Trinomials 367 5.7 Factoring Binomials Including Sum and Difference of Cubes 379 Problem Recognition Exercises: Factoring Summary 389 vii 5.8 Solving Equations and Applications by Factoring 391 Group Activity: Investigating Pascal’s Triangle 405 Chapter Summary 406 Chapter Review Exercises 411 Chapter Test 415 Chapters 1–5 Cumulative Review Exercises 416 Chapter Rational Expressions and Rational Equations 6.1 Rational Expressions and Rational Functions 420 6.2 Multiplication and Division of Rational Expressions 6.3 Addition and Subtraction of Rational Expressions 6.4 Complex Fractions 419 430 435 444 Problem Recognition Exercises: Operations on Rational Expressions 451 6.5 Solving Rational Equations 451 Problem Recognition Exercises: Rational Equations Versus Expressions 459 6.6 Applications of Rational Equations and Proportions 6.7 Variation 460 470 Group Activity: Computing the Future Value of an Investment Chapter Summary 479 480 Chapter Review Exercises 485 Chapter Test 488 Chapters 1–6 Cumulative Review Exercises Chapter Radicals and Complex Numbers 7.1 489 491 Definition of an nth Root 492 7.2 Rational Exponents 504 7.3 Simplifying Radical Expressions 511 7.4 Addition and Subtraction of Radicals 519 7.5 Multiplication of Radicals 524 Problem Recognition Exercises: Simplifying Radical Expressions 532 7.6 Division of Radicals and Rationalization 532 7.7 Radical Equations and Applications 542 7.8 Complex Numbers 552 Group Activity: Margin of Error of Survey Results Chapter Summary 563 Chapter Review Exercises 569 Chapter Test 572 Chapters 1–7 Cumulative Review Exercises viii 573 561 Chapter Quadratic Equations, Functions, and Inequalities 575 8.1 Square Root Property and Completing the Square 576 8.2 Quadratic Formula and Applications 8.3 Equations in Quadratic Form 585 599 Problem Recognition Exercises: Identifying and Solving Equations 605 8.4 Graphs of Quadratic Functions 606 8.5 Vertex of a Parabola: Applications and Modeling 8.6 Polynomial and Rational Inequalities 619 629 Problem Recognition Exercises: Recognizing Equations and Inequalities 640 Group Activity: Creating a Quadratic Model of the Form y ؍a(x ؊ h)2 ؉ k 641 Chapter Summary 642 Chapter Review Exercises Chapter Test 647 650 Chapters 1–8 Cumulative Review Exercises Chapter 652 Exponential and Logarithmic Functions 655 9.1 Inverse Functions 656 9.2 Exponential Functions 665 9.3 Logarithmic Functions 675 Problem Recognition Exercises: Identifying Graphs of Functions 688 9.4 Properties of Logarithms 689 9.5 The Irrational Number e 697 Problem Recognition Exercises: Logarithmic and Exponential Forms 709 9.6 Exponential Equations and Applications 710 9.7 Logarithmic Equations and Applications 719 Group Activity: Creating a Population Model Chapter Summary 727 Chapter Review Exercises Chapter Test 726 731 735 Chapters 1–9 Cumulative Review Exercises 737 ix SA-51 Student Answer Appendix 19 a y $ x2 1 25 24 23 22 21 21 13 x 22 23 24 25 b The parabola y ϭ x2 ϩ would be drawn as a dashed curve 21 1x Ϫ 32 ϩ 1y ϩ 42 Յ 625 23 25 y y 5 4 2x y $ x # y2 1 x 25 24 23 22 21 21 22 23 23 24 24 25 29 x y51 2 x 25 24 23 22 21 21 22 23 22 23 25 33 y 25 24 23 22 21 21 x y2 $ 4 x x 25 37 y y # ln x y y 5x x x 22 1 x x x y 45 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 x2 y2 25 51 (23, 0) y 47 No Solution 49 51 y 25 24 23 22 21 21 22 23 24 25 x y x2 4y2 36 (3, 0) (26, 0) x (0, 3) 26 25 24 23 22 21 21 22 33 Center: (0, 2) (5, 1) x 22 21 21 22 23 (3, 23) (7, 23) (5, 23) 24 25 26 27 28 x 22 Chapter 10 Review Exercises, pp 793–796 110 189 x ϭ or x ϭ Ϫ1 x ϭ Ϫ3 Center (12, 3); r ϭ Center 1Ϫ7, 52; r ϭ Center 1Ϫ3, Ϫ82; r ϭ 215 Center 11, Ϫ62; r ϭ 412 a x2 ϩ y2 ϭ 64 b 1x Ϫ 82 ϩ 1y Ϫ 82 ϭ 64 10 1x ϩ 62 ϩ 1y Ϫ 52 ϭ 10 (25, 2) (0, 2) y2 (22, 0) (5, 2) x 23 24 25 (5, 27) y x (0, 5) 25 24 23 22 21 21 (0, 21) 22 34 Horizontal 35 Vertical 36 Vertical 37 Horizontal 38 39 x 23 (0, 23) 24 25 26 y x (6, 0) y y 25 24 23 22 21 21 31 (0, 5) 32 Center: (5, Ϫ3) 25 24 23 22 21 21 23 24 25 (0, 25) 26 x x y 5 1 29 y ϭ Ϫ2 ax ϩ b ϩ ; vertex: aϪ , b ; 2 2 26 25 24 23 22 21 21 22 25 26 y ϭ 1x Ϫ 32 Ϫ 4; vertex: (3, Ϫ4); axis of symmetry: xϭ3 27 x ϭ 1y ϩ 22 Ϫ 2; vertex: (Ϫ2, Ϫ2); axis of symmetry: y ϭ Ϫ2 28 x ϭ Ϫ4 ay Ϫ b ϩ 1; vertex: a1, b ; 2 30 25 24 23 22 21 21 22 23 24 25 y50 24 25 axis of symmetry: x ϭ Ϫ y 25 24 23 22 21 21 22 23 24 x y (x 2)2 25 24 23 22 21 21 22 23 25 41 y y (21, 0) x 5 2 x 2y 21 (0, 0) 24 25 25 axis of symmetry: y ϭ 25 24 23 22 21 21 22 23 24 24 25 25 24 23 22 21 (22, 0) 21 22 23 24 x50 25 23 24 25 x 25 24 23 22 21 21 22 24 25 24 23 22 21 21 22 23 23 x x 16y2 # 16 9x2 y2 22 25 24 23 22 21 21 22 23 y 1 4 y y52 x 24 25 24 24 1 25 24 23 22 21 21 23 x 2(y 1)2 24 25 y 5 (0, 1) 25 24 23 22 21 21 22 4 43 25 y (x 1)2 (y 2)2 39 y y 22 27 35 15 18 19 20 21 22 25 24 23 22 21 21 31 12 ax Ϫ b ϩ 1y Ϫ 22 ϭ 2 49 2 14 x ϩ y2 ϭ 1x Ϫ 32 ϩ ay Ϫ b ϭ 16 1Ϫ4, Ϫ22 17 1Ϫ1, 82 x2 ϩ y Ϫ 22 ϭ Vertical axis of symmetry; parabola opens downward Horizontal axis of symmetry; parabola opens right Horizontal axis of symmetry; parabola opens left Vertical axis of symmetry; parabola opens upward 23 11 1x ϩ 22 ϩ 1y ϩ 82 ϭ y y 6 5 (0, 4) 26 25 24 23 22 21 21 22 23 (2, 0) y2 x2 16 x 24 25 26 40 Hyperbola 41 Ellipse 26 25 24 23 22 21 21 22 23 ( 0, 24) 24 25 26 x SA-52 Student Answer Appendix 5 Center: a , Ϫ b; r ϭ 42 Ellipse 43 Hyperbola 44 a Line and parabola 45 a Line and parabola b b y (24, 11) y 12 6 (52 , 54 ) 25 24 23 22 21 22 x (3, 21) 26 25 24 23 22 21 23 24 y 18 15 12 (25, 15) 10 4 (22, 1) x 29 28 x522 212 5 c a , b, 1Ϫ4, 112 46 a Circle and line b c 1Ϫ5, 152, 13, Ϫ12 47 a Circle and line b y y (0, 3) 25 24 23 22 21 21 x 25 2423 22 21 21 (125, 95) 22 23 50 52 53 54 y x 16 81 (165 , 2125 ) 56 x 10 25 24 23 22 21 21 22 26 28 23 24 210 25 57 y 2 (x 3) (y 1) $ v 25 24 23 22 21 21 22 x 10 y2 x2 # 25 24 23 22 21 21 22 y (x 1)2 23 24 x 61 4 x 25 24 23 22 21 21 23 24 25 26 25 24 23 22 21 21 22 23 24 x 2(y 2) 25 26 x x y 5 x 25 24 23 22 21 21 22 y $2 13 x 123 24 25 17 y 5y 3x 15 y c (3, 2) a 1Ϫ5, 02, 10, 32 y y52 x 27 26 25 24 23 22 21 21 22 23 24 25 x b m ϭ 5 x y 25 24 23 22 21 21 22 23 24 25 All real numbers ) 10 a , ϱ b 10 3 The integers are 10 and 15 x 22 5 25 24 23 22 21 21 22 23 24 25 x Chapter 10 Test, pp 796–797 x (0, 27) Chapters 1–10 Cumulative Review Exercises, pp 798–799 23 22 23 24 1 29 28 27 26 25 24 23 22 21 21 y x 25 24 23 22 21 21 3 x#y 11 16 25 24 23 22 21 21 22 23 24 25 Ϫ7 24 25 y 25 25 24 23 22 21 21 22 23 25 11 a 1Ϫ3, 0210, 42; ii b No solution; i 12 The addition method can be used if the equations have corresponding like terms 13 12, 021Ϫ2, 02 14 15 x2 y2 Ϫ ϭ1 22 23 2 22 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ6 (x 2)2 (y 1)2 # x 25 24 23 22 21 21 59 (4, 0) (24, 2) y (1, 3) (24, 3) y 24 25 y (24, 4) (29, 3) y y 25 60 27 23 24 58 y 6 (24, 0) 210 28 26 24 22 22 24 x 27 26 25 24 23 22 21 21 22 23 24 25 26 y2 x2 25 ,1 110 Center: 10, 22; r ϭ a 15 b x2 ϩ 1y Ϫ 42 ϭ The center is the midpoint (3.8, 1.95) (0, 7) y 2 y (x 2)2 11 x 16 12 12 ,Ϫ b c 10, Ϫ42, a , Ϫ b 5 5 13 16 14 10, Ϫ22, a , b 49 aϪ , b, 1Ϫ5, Ϫ12 9 5 18, 42, 12, Ϫ22 51 12, 42, 1Ϫ2, 42 , 12, 22, 1Ϫ12, 22 13, 12, 13, Ϫ12, 1Ϫ3, 12, 1Ϫ3, Ϫ12 16, 52, 1Ϫ6, 52 , 16, Ϫ52, 1Ϫ6, Ϫ52 55 y y 24 (0, 24) 25 10 23 24 25 26 c 10, 32, a 48 x2 y2 51 16 49 22 23 24 25 26 25 24 23 22 21 21 22 26 26 5 x Student Answer Appendix 3 slope: ; y-intercept: a0, Ϫ b There are 12 dimes and quarters 12, Ϫ3, 12 a , b (6, 3) 2 10 f 102 ϭ Ϫ12; f 1Ϫ12 ϭ Ϫ16; f 122 ϭ Ϫ10; f 142 ϭ Ϫ16 11 g122 ϭ 5; g182 ϭ Ϫ1; g132 ϭ 0; g1Ϫ52 ϭ 12 z ϭ 32 13 1g ؠf 21x2 ϭ x ϩ 7; x Ն Ϫ1 14 a Ϫ5 b 1x ϩ 121x2 ϩ 12; Ϫ5 c They are the same 15 1x ϩ y2 1x Ϫ y Ϫ 62 aϪ1 17 6, 18 aϩ2 16 x3 Ϫ 2x2 Ϫ 7x Ϫ Ϫx2 Ϫ x Ϫ 19 20 0, Ϫ1 1x ϩ 321x Ϫ 22 21 a No solution b Ϫ11 22 Ϫ30 ϩ 24i 12 15 ϩ i 23 24 17 m 41 41 25 a 17.6 ft; 39.6 ft; 70.4 ft b sec 13 1 Ϯ i 26 Ϫ , 27 Ϯ 211 28 1Ϫ5, Ϫ362 10 10 29 a 1Ϫ3, 02, 11, 02 b (0, 3) c 1Ϫ1, 42 y (0, 3) ( 1, 4) ( 3, 0) (1, 0) 1 3 x 31 aϪϱ, 30 (Ϫ1, 19) 33 d ´ c , ϱb 2 34 hϪ1 1x2 ϭ x ϩ 36 32 log8 32 ϭ 53 35 x2 ϩ 1y Ϫ 52 ϭ 16 y 25 24 23 22 21 21 22 23 24 25 37 aϪ , 0b 39 y x , 5y x 38 10, Ϫ42 4 3 2 25 24 23 22 21 21 22 23 24 25 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 17 19 False 21 True 23 6! ϭ ؒ (5 ؒ ؒ ؒ ؒ 1) ϭ ؒ 5! 25 1680 27 29 56 31 33 m11 ϩ 11m10n ϩ 55m9n2 35 u24 Ϫ 12u22v ϩ 66u20v2 37 terms 39 s6 ϩ 6s5t ϩ 15s4t ϩ 20s3t ϩ 15s2t ϩ 6st ϩ t 41 b3 Ϫ 9b2 ϩ 27b Ϫ 27 43 16x4 ϩ 32x3y ϩ 24x2y2 ϩ 8xy3 ϩ y4 45 c14 Ϫ 7c12d ϩ 21c10d Ϫ 35c8d ϩ 35c6d Ϫ 21c4d ϩ 7c 2d Ϫ d 5 5 47 a Ϫ a4b ϩ a3b2 Ϫ a2b3 ϩ ab4 Ϫ b5 32 16 2 49 x4 ϩ 16x3y ϩ 96x2y2 ϩ 256xy3 ϩ 256y4 51 Ϫ462m6n5 53 495u16v4 55 g9 Section A.2 Practice Exercises, pp A15–A18 a determinant; ad Ϫ bc b minor c a2; ` b1 c1 ` b2 c2 16 11 26 13 15 46 17 a 30 b 30 19 Choosing the row or column with the most zero elements simplifies the arithmetic when evaluating a determinant 21 12 23 Ϫ15 25 27 8a Ϫ 2b 29 4x Ϫ 3y ϩ 6z 31 33 D ϭ 16; Dx ϭ Ϫ63; Dy ϭ 66 35 (2, Ϫ1) 23 37 (4, Ϫ1) 39 a , b 13 13 41 Cramer’s rule does not apply when the determinant D ϭ 43 No solution; Inconsistent system 45 (0, 0) 47 Infinitely many solutions; 1x, y2 x ϩ 5y ϭ 36 ; Dependent equations 49 x ϭ 1 16 51 z ϭ 53 y ϭ 55 (3, 2, 1) 41 57 Cramer’s rule does not apply 59 x ϭ Ϫ19 61 w ϭ 63 36 65 a b Ϫ2 c x ϭ Ϫ1 67 The measures are 37.5° and 52.5° 69 40 iPods, 15 iPads, and 20 iPhones were sold 71 There were 560 women and 440 men in the survey Section A.3 Practice Exercises, pp A23–A26 y 40 SA-53 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 Additional Topic Appendix Section A.1 Practice Exercises, pp A6–A7 a a2 ϩ 2ab ϩ b2; a3 ϩ 3a2b ϩ 3ab2 ϩ b3; binomial b n1n Ϫ 12 1n Ϫ 22 p 122112; factorial c 6; 2; 1; d binomial e Pascal’s a3 ϩ 3a2b ϩ 3ab2 ϩ b3 ϩ 4g ϩ 6g2 ϩ 4g3 ϩ g4 7 p ϩ 7p q ϩ 21p q ϩ 35p4q6 ϩ 35p3q8 ϩ 21p2q10 ϩ 7pq12 ϩ q14 s5 ϩ 5s4t ϩ 10s3t2 ϩ 10s2t3 ϩ 5st4 ϩ t5 11 625 Ϫ 500u3 ϩ 150u6 Ϫ 20u9 ϩ u12 13 x6 Ϫ 12x4 ϩ 48x2 Ϫ 64 15 120 a infinite; finite b terms; nth c alternating d series e summation f index; (3)2 ϩ (4)2 ϩ (5)2 1 1, Ϫ1, Ϫ3, Ϫ5, Ϫ7 0, 1, 12, 13 , , , , 7 11 0, , Ϫ , 11 1, Ϫ4, 13 , , , 5 15 0, 6, 24, 60, 120, 210 17 Ϫ1, 2, Ϫ3, 19 When n is odd, the term is positive When n is even, the term is negative n 21 an ϭ 3n 23 an ϭ 2n ϩ 25 an ϭ nϩ1 27 an ϭ (Ϫ1)n 29 an ϭ (Ϫ1)nϩ1 3n 31 an ϭ n 33 $30.00, $30.90, $31.83, $32.78 1 31 35 16, 8, 4, 2, 1, , , 1grams2 37 90 39 16 41 30 43 10 45 73 12 47 38 49 Ϫ1 SA-54 Student Answer Appendix 51 55 53 a n nϭ1 59 a 1Ϫ12 kϩ1 kϭ1 3k 55 a 57 a j 61 a xn 63 15.4 g iϭ1 jϭ1 nϭ1 65 4.8 g 67 Ϫ3, 2, 7, 12, 17 71 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 69 5, 21, 85, 341, 1365 Section A.4 Practice Exercises, pp A35–A38 a arithmetic b difference c a1 ϩ 1n Ϫ 12d; d n d series e 1a1 ϩ an f geometric g ratio a1 11 Ϫ rn a1 nϪ1 h a1r i j ;1 1Ϫr 1Ϫr Ϫ3 Ϫ2 3, 8, 13, 18, 23 11 2, , 3, , 13 2, Ϫ2, Ϫ6, Ϫ10, Ϫ14 2 15 an ϭ Ϫ5 ϩ 5n 17 an ϭ Ϫ2n 19 an ϭ ϩ n 2 21 an ϭ 25 Ϫ 4n 23 an ϭ Ϫ14 ϩ 6n 25 a6 ϭ 17 27 a9 ϭ 47 29 a7 ϭ Ϫ30 31 a11 ϭ Ϫ48 33 19 35 22 37 23 39 11 41 a1 ϭ Ϫ2, a2 ϭ Ϫ5 43 670 45 290 47 Ϫ15 49 95 51 924 53 300 55 Ϫ210 57 5050 59 980 seats; $14,700 61 A sequence is geometric if the ratio between each term and the preceding term is constant 63 65 Ϫ3 67 69 Ϫ4, 4, Ϫ4, 4, Ϫ4 1 71 8, 2, , , 73 2, Ϫ6, 18, Ϫ54, 162 32 nϪ1 75 an ϭ 2132 77 an ϭ Ϫ61Ϫ22 nϪ1 79 an ϭ 16 nϪ1 a b 87 Ϫ10 89 81 Ϫ 91 128 93 83 Ϫ 1562 125 81 85 135 95 Ϫ 11 2059 3124 99 101 Ϫ172 27 729 103 a $1050.00, $1102.50, $1157.63, $1215.51 b a10 ϭ $1628.89; a20 ϭ $2653.30; a40 ϭ $7039.99 1 105 r ϭ ; sum is 107 r ϭ Ϫ ; sum is 5 109 r ϭ Ϫ ; sum is Ϫ 111 r ϭ Ϫ ; sum does not exist 113 $800 million 115 28 ft 117 a $1,429,348 b $1,505,828 c $76,480 97 www.downloadslide.net Photo Credits Photo Credits: Page 2: © Getty /Blend Images RF; p 3: © Photodisc/Getty RF; p 61: © Corbis RF; p 64: © Rob Melnychuk/Getty RF; p 66: © John Lund/Drew Kelly/Blend Images RF; p 69: © Brand X Pictures/PunchStock; p 81: © Creatas Images/MasterFile RF; p 92: © Image Source RF; p 93: © Don Hammond/Design Pics RF; p 108: © BananaStock/Punchstock RF; p 122: © Keith Brofsky/Getty RF; p 130: © Corbis RF; p 171: © PhotoDisc/ Getty RF; p 172: 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RF; p 722: Photo by M Celebi, U.S Geological Survey; p 759(top): © Getty/Photodisc RF; p 759(bottom): © Royalty-Free/Corbis C-1 Application Index Application Index Biology/Health/Life Sciences Adenosine deaminase levels in humans, 122 Age range for average height of boys, 81 Amount of medicine prescribed by weight of patient, 476 Average weight for boys/girls based on age, 151 Bacteria population as function of time, 627, 674, 717, 736, 738 Bison population in Wyoming, 467 Blood pressure, normal range of, 15 Calcium intake program, 316 Calories in candy, 467 Cholesterol level ranges, 120, 461 Epidemic deaths as function of time, 708 Fat amount in foods, 484 Feasible cage configurations at animal shelter, 278 Femur length vs height in women, 190, 193–194 Hemoglobin range in humans, 92 Length of human pregnancy, 88 Life expectancy vs year of birth, 186 Longevity of animals, by species, 191 Maximum heart rate vs age, 120, 197 Normal level of TSH for adults, 88 Number of manatees in Florida, 467 pH level in ammonia, 682 pH level in antacid tablet, 733 pH level in blood, 686 pH level in shampoo, 682 pH ranges in food substances, 15, 733 Pollution in atmosphere, 477 Radioactive decay, 703 Ratio of cats to dogs in shelter, 461 Sodium content of foods, from total sodium intakes, 290–291 Water level retention in pond, 78–79 Weight vs age for children, 151 White blood cell range in humans, 92 Business and Economics Airline flights as function of passenger volume, 653 Amount of each nut in nut mixture, from relative amounts, 294 Cell phone subscription trends, 150 Commission, 64 Cost of airline operation, 550 I-1 Cost of cab ride per mile, 138 Cost of running business, 184 Cost per item as function of number produced, 403, 477 Cost to package product, 627 Currency exchange rates, 487 Depreciation, 130–131, 139, 734 Drinks, price of related to number sold, 174 Earnings needed to meet average salary level, 81 Hot dogs, price of related to number sold, 174 Interest compound, 584, 699, 705, 717, 733, 737, A–37 simple, 58, 59, 64–65, 66, 112, 269, 474, 490 Markups/markdowns, 58–59, 64, 66 Median weekly salary of individual with degree, 66 Minimum wage trends, 167 Mixing foods for resale, 66, 294 Mixing money among investments, 59–60 Monthly salary, 78 Number of fish, from weight of total catch, 467 Orders processed per day by each worker, 316 Profit as function of number of items produced, 81, 228, 337, 584 Profit as function of number of items sold, 81, 477, 649 Ratio of men to women in accounting firm, 467 Salary plan, 665–666 Salary with commissions, vs sales, 138, 171, 186 Salary with tips, vs number of tables served, 234 Sales as function of advertising expenditures, 687 Sales by each salesperson, from relative totals, 290 Time required for each person working alone to finish job, 469, 470 Time required for two workers to finish job, 285, 469 Tourist revenue, A–38 Construction and Design Bridge support design, 618 Bulge height in heat-expansion of bridge, 570 Cell tower location, 748, 783 Cost to carpet a room, 229 Dimensions of fenced area, 628 Elevator passenger capacity, 122 Fencing needed to enclose area, 71, 285, 518, 628 Length of roof, from structure dimensions, 572 Length of screw, 121 Length of tower guy wires, 571 Margin of error in measurement, 108, 122 Maximum enclosable area with given amount of fence, 71 Outdoor fountain water projection, 759 Production of two models of desks, 286 Pump performance, from relative performance, 314, 469 Slope of hill, 148 Slope of ladder, 142, 148, 402 Slope of leaning telephone pole, 417 Slope of roof, 142, 148 Slope of treadmill, 148 Stained glass window design, 599, 793 Strength of beam as function of dimensions, 478 Time required for two workers to finish job, 285, 465, 489 Time required to fill tank, from pipe size, 488 Window film needed for project, 25 Consumer Applications Car rental cost vs mileage, 272 Cellular phone charges, 167 Cellular phone subscriptions in U.S., 150 City and highway miles driven, 263 Coins by type, from relative number, 272, 798 Cook time vs weight for turkey, 551 Cost of bottle of laundry detergent by size, 467 Cost of cab ride, 138 Cost of car rental, 170 Cost of items, from money spent, 269–270, 294, 467 Cost of nights in motel, 256 Cost to rent apartment, 256 Dog run dimensions, 71, 285 Gas mileage, 467 Gas mileage as function of speed, 627 Gas required to complete trip, 29 Home value, A–37 Application Index Income after taxes, 118 Income tax owed, 118 Lengths of rope needed for rope trick, 118 Loan amount borrowed, from interest paid, 112 Measurement error in contents of food packaging, 108 Median price of new home, 66 Mixing products, 65, 294 Number of items, from total number, A–18 Number of kilowatt-hours used, 54 Number of people served by turkey, by weight, 476 Picture frame dimensions, 71 Property tax trends, 171 Rabbit pen dimensions, 72 Rental costs, 311 Sales tax, 64 Satellite dish depth, 759 Spending on snacks at movie, 264 Telephone company charges compared, 272, 312 Television audience viewing American Idol, 121 Textbook cost vs size, 168 Tire wear vs tire pressure, 627 Video rental costs compared, 317 Weight of car vs gas mileage, 194 Distance/Rate/Time Airplane landing distance vs speed, 647 Altitude gain over horizontal distance, 149 Average speed of boats, 66 Average speed of bus and train, 464 Average speed of car and train, 464 Average speed of car overtaking bus, 469 Average speed of racecar, 72 Average speed of walker, 469 Car braking distance vs speed, 548, 597 Distance between cities, from map, 467, 489, 490, 588 Distance between vehicles/walkers traveling apart, 498, 502, 588 Distance between vertices in arch, 768 Distance from Earth to celestial objects, 325, 328 Distance of lightning strike, calculating, 172 Distance to location, 121, 402, 749 Distance traveled, from time and speeds, 209, 799 Gas remaining vs time, 131 Height vs horizontal distance, 338, 415, 518, 627, 651, 768 Height vs time in free fall, 209, 399, 551 Height vs velocity, 550, 589, 597, 623, 652 Maximum height of launched object, 623, 627, 649 Number of stacked pennies, 325 Speed of airplane discounting wind speed, 267–268, 271, 312 Speed of airplane trips, 65 Speed of airplane with tailwind, 267 Speed of bicyclist against wind, 292, 469, 489 Speed of bicyclist discounting wind speed, 292, 489 Speed of boat as function of current, 269, 271, 272, 469 Speed of boat discounting water speed, 269, 271, 272, 469 Speed of car as function of time, 490 Speed of cars/travelers, from distance of separation, 65, 118, 469 Speed of cross-country skiers, 463 Speed of driving and walking, 488 Speed of driving in clear weather, 61 Speed of hiker, 61–62, 65 Speed of motorist in rainstorm, 468 Speed of moving sidewalk, 271, 469 Speed of runner and biker, 463 Speed of travelers, from relative speeds, 518 Speed of triathlon bicyclist, 271 Speed of triathlon runner, 469 Speed of walking and cycling, 469 Speed of walking on nonmoving ground, 271, 469 Speed vs time for vehicle, 469 Stopping distance vs speed, 477 Time of car races, 72 Time of height after launch, 399–400, 403, 414 Time required for object to fall, 399, 541, 548, 581–582, 585 Time required for pendulum swing, 478, 489, 541, 550 Time required for travelers to meet, 118 Time required for trip, 121 Time required for two workers to finish job, 285, 469 Time required to overtake traveler, 737 Time required to reach investment goal, 73, 81 Wind speed, 267–268, 292, 312 Education and School Associate degrees conferred, trends in, 174 College attendance by males, trends in, 118 Dormitory fee trends, 338 Grade average relating to final exam, 93, 221 Grades needed to make desired average, 78, 81, 119, 417 I-2 Median weekly salary of individual with degree, 66 Number of credit hours taken, 54 Ratio of children to adults, 467 Ratio of passing to failing students, 461 Ratio of single men to single women in college, 467 Retention of material as function of time, 682–683, 734, 736 Study time availability analysis, 277–279 Study time vs grade, analysis of, 175, 209 Teaching methods of vocabulary, 687 Textbook cost vs size, 168 Environment/Earth Science/Geography Annual loss due to earthquakes in California, 328 Atmospheric pressure as function of altitude, 737 Average low/high temperature, 28, 93 Average temperature by city, 662 Conversion of degrees Celsius to/from Fahrenheit, 28 Hurricane intensity, 12 pH level of rainwater, 682 Precipitation relating to month of year, 197 Richter scale for earthquakes, 697, 722–723, 724, 735 Slope of altitude increase of airplane, 149 Snowfall rate and totals, 166 Temperature vs altitude, 139 Visibility vs altitude, 488 Water level vs days of drought, 173 Weak earthquakes, 783 Yearly rainfall for two cities, 238 Gardening and Landscaping Acreage for planting fruit trees, 313–314 Area of walkway and garden as function of walkway width, 347, 412 Dimensions of area enclosable with available fencing/edging, 71, 285 Dimensions of garden, from perimeter, 71, 113 Edging around tree, 72 Fertilizer mixtures, 65 Height of tree, from shadow cast, 462 Perimeter of garden as function of side length, 337, 524 Radius of garden, from area, 650 Geometry Angle measure of supplementary angles, 271 Angle measures of complementary angles, 271, 312, 316 I-3 Application Index Angle measures of triangle, 271, 289–290, 293, 314 Area as function of side length, 25, 37, 344, 347, 412, 478, 510 Area of region, 348, 389, 411, 477 Dimensions of region, from area, 402, 584, 650 Dimensions of region, from perimeter, 66–67, 404 Dimensions of region, from volume, 584, 597 Height of triangle, 402 Length of triangle side, 71, 121, 314, 397, 402, 468, 518, 584, 597 Measurement of angles of triangle, 72 Perimeter as function of side length, 272, 293, 333, 337, 412, 489 Pythagorean Theorem, 397, 497 Radius of circle, from area, 68–69, 403, 404, 650 Radius of sphere, from volume, 510 Volume as function of side length, 344, 347–348, 414, 547–548 Volume of figure, 29 Investment Amount in account as function of time, 674, 718 Amount invested, from return, 58, 118, 121, 266, 270, 272, 294, 312, 366, 652 Annual rate of return, 510 Bond yield, 473 Interest compound, 584, 699, 705, 717, A–37 simple, 58, 59, 64–65, 270, 474, 477, 490 Savings deposits per month, 328 Savings totals with monthly deposits, 166 Sequence of payments, A–24–A–25 Time required to reach investment goal, 73, 81, 708 Yearly salary with annual raise, A–38 Politics and Public Policy Child support due vs child support paid, 228, 338 Economic bailout, 325 Margin of error in polls, 108 Minimum wage trends, 167 Number of representatives in House of Representatives, 662 Polling company estimates, 92 Depth of water, from wave velocity, 571 Distance from Earth to celestial objects, 325, 328 Distance of lightning strike, calculating, 172 Electrical current vs voltage and resistance, 477, 478 Electrical resistance vs wire size, 477 Force on spring vs distance stretched, 172, 217, 478, 488 Force required to stretch spring, 172, 217 Frequency vs length in vibrating string, 477 Height vs time in free fall, 209, 399 High-energy particle accelerator, 327 Intensity of light as function of distance, 477 Kinetic energy, 474 Magnitude of star, 724 Margin of error in measurement, 105 Mixing solutions, 60–61, 65, 118, 121, 238, 265, 269, 270, 272, 312, 316, 417, 467, 737 Pendulum period vs length, 478, 489, 541, 550 pH measurement, 682, 686, 724–725 Population of bacteria vs time, 338 Radar location, 748 Radioactive decay, 669–670, 673, 714–715, 717–718, 732, 734 Sequence of heights in bouncing ball, A–21, A–34 Sequence of pendulum swings, A–21 Sequence representing bacteria population, A–25 Sequence representing radioactive decay, A–25 Sound intensity in decibels, 473–474, 697, 724 Stopping distance of car, 477 Thickness of metal measured by machine, 105 Thickness of wood measured by machine, 92 Velocity of falling object, 217 Vertical distance traveled by ball, A–38 Volume of gas as function of temperature and pressure, 738 Weight of ball as function of radius, 478 Year of first man or woman in space, 195 Science and Technology Sports/Exercise/ Entertainment Acceleration of mass influenced by force, 73 Atoms in substance, number of, 328 Conversion of degrees Celsius to/from Fahrenheit, 28 Amount spent on video games, 172 Array of dominoes, A–36 Daytona 500 car race speed, 72 Dig for buried treasure, 312 Dimensions of court, 71, 396 Distance biker rides, 237 Distance from home plate to second base, 518 Golf holes by par type, 294 Indianapolis 500 car race speed, 72 Medicine ball weight, 477 Men’s pole vault height, trends in, 173 Number of commercials, A–18 Points scored, by type of shot, 272 Points scored, from relative scores, 416 Points scored during basketball season, 652 Projected yardage gain from number of passes, 487 Revenue from theater, A–36 Revolution of bicycle wheels, 229 Speed of faster car, 187 Speed of racing canoe, 473 Spending on snacks at amusement park, 264 Tickets sold by type, 272, 312 Vertical distance traveled by bungee jumper, A–38 Water needed by hiker, 473 Women’s swimming times, trends in, 168–169 Women’s track and field records, trends in, 197 Workout times, A–18 Yards rushed vs margin of victory, 184 Statistics and Demographics Alcohol-related deaths, trends in, 118 Bottled water consumption, 235, 411 Child support due vs child support paid, 228, 338 Fatality rate for drivers, 597 Males employed full-time in U.S., 147 Margin of error in polls, 108 Miles driven per year, trends in, 172 Number living in poverty, 195 Number of representatives in House of Representatives, 191 Numbers, from relative size, 121 Per capita income in U.S., 181 Percentage of males who smoke, 334 Population density, 147, 328 Population density trends, 197, 416, 651 Population growth in U.S., 150 Population growth trends, 78, 229, 334, 338, 411, 416, 648, 671, 674, 714, 717, 736 Prisoners in federal or state facilities, 174 Ratio of male to female police officers, 461 Soft drink consumption per capita, trends in, 237 Weekly earnings for women, 221 Year of statehood, 195 Subject Index A Abscissa See x-coordinate Absolute value bars, 496 Absolute value equations explanation of, 93, 116 methods to solve, 94–97, 100, 116 Absolute value expressions, 105 Absolute value functions, 213 Absolute value inequalities explanation of, 99, 116 method to solve, 100–102, 116 test point method to solve, 102–104, 117 Absolute values applications of, 105 equality of, 97 of real numbers, 16, 17 translating to/from words, 105 AC-method, to factor trinomials, 367–369, 408 Addition associative property of, 31 commutative property of, 31 of complex numbers, 556–557 distributive property of multiplication over, 31, 39 of functions, 223–224 identity property of, 31 inverse property of, 31 of like terms, 32, 33, 406 of polynomials, 330, 406 of radicals, 519–521 of rational expressions, 435, 438–441, 482 of real numbers, 17–18, 38 Addition method explanation of, 256 to solve nonlinear systems of equations, 773–775 to solve systems of linear equations in two variables, 256–259, 307 Addition property of equality, 45 Addition property of inequality, 74 Algebraic expressions explanation of, 30 exponents to simplify, 320–323 simplification of, 33–34, 39 terms of, 30 Algorithms, division, 351 Alternating sequences, A–19, A–20 Angles complementary, 67–68, 113 supplementary, 113 in triangles, 289–290 Applications of absolute value, 105 of compound inequalities, 88–89 of consecutive integers, 57 distance, rate, and time, 61–62, 463, 464 of exponential equations, 714–715 of exponential functions, 669–671 of geometry, 66–68, 113, 268, 547–548 of inequalities, 78–79 of linear equations in one variable, 55–62, 112 of linear equations in three variables, 289–291 of linear equations in two variables, 147, 165–169, 181 of linear inequalities, 78–79 of logarithmic equations, 722–723 of logarithmic functions, 682–683, 703 of mixtures, 60–61, 265 of percents and rates, 58–59 of polynomial functions, 333–334 of principal and interest, 59–60, 266 of product of polynomials, 343–344 of Pythagorean theorem, 497, 498 of quadratic equations, 396–397 of quadratic formula, 588–589 of quadratic functions, 399–400, 589, 623 of radical equations, 547–548 of radical functions, 548 of rational exponents, 508 of relations, 193–194 of scientific notation, 325 of sequences, A–21 of series, A–34 of similar triangles, 462 of slope, 142, 147 of systems of linear equations in three variables, 289–291 of systems of linear equations in two variables, 264–268, 307 of uniform motion, 267–268 of variation, 472–474 Approximation of exponential expressions, 666–667 of nth roots, 495 Area of circle, 581 formulas for, 113 Argument, of logarithmic expressions, 675 Arithmetic sequences explanation of, A–26 – A–27 finding number of terms in, A–28 finding specified term of, A–28 nth term of, A–27 Arithmetic series explanation of, A–28 – A–29 sum of, A–29 – A–30 Associative property of addition, 31 Associative property of multiplication, 31 Asymptotes, 212 Augmented matrices, 296–298, 310 Axis horizontal transverse, 763, 790 transverse, 762–763 vertical transverse, 763, 790 Axis of symmetry explanation of, 751 of parabola, 751–754, 789 B b0, 320 Base of exponential expression, 22, 38 of logarithmic expressions, 675 Binomial expansion binomial theorem for, A–3 – A–5 explanation of, A–1 – A–2 factorial notation and, A–2 – A–3 finding specific term in, A–5 Pascal’s triangle and, A–2 Binomial factors, 361 Binomials See also Polynomials difference of squares to factor, 379–382, 409 explanation of, 329, 406 factoring, 361, 379–386, 409–410 (See also Factors/factoring) of form x6 Ϫ y6, 385–386 multiplication of, 341–343 square of, 342, 343, 526, A–1 sum and difference of cubes to factor, 382–384 summary of factoring methods for, 384–385 Binomial theorem applications of, A–4 – A–5 explanation of, A–3 – A–4 bϪn, 320, 321 Boundary points, 630–633, 646 Braces, 23 Brackets, 10, 11, 23 C Calculators See Graphing calculators Center, of circle, 741 Change-of-base formula explanation of, 701–702 on graphing calculators, 702 Circles explanation of, 741 formula for area of, 581 graphs of, 741–743 radius of, 741 standard equation of, 741, 788 Clearing decimals, 50 Clearing fractions, 48, 49, 452, 453 Clearing parentheses, 34 Coded messages, 229–230 Coefficients, 30, 39 Column matrices, 296 Commission, 58 Common difference, of arithmetic sequence, A–27 Common logarithmic functions application of, 682 explanation of, 677–678, 728 I-4 I-5 Subject Index Common ratio, of geometric sequence, A–30, A–31 Commutative property of addition, 31 Commutative property of multiplication, 31 Complementary angles, 67–68, 113 Completing the square explanation of, 578–579, 645 to graph parabolas, 752–753, 755 to solve quadratic equations, 579–581 vertex formula derived by, 621 to write equation of circle, 743 Complex conjugates, 555, 568 Complex fractions explanation of, 444, 483 methods to simplify, 444–448 Complex numbers addition of, 556–557 division of, 558 explanation of, 555, 568 multiplication of, 557, 558 real and imaginary parts of, 556 simplification of, 559 subtraction of, 556–557 Composition, of functions, 224–225, 233 Compound inequalities applications of, 88–89 explanation of, 84, 115 of form a Ͻ x Ͻ b, 86–87 joined by word and, 84–85 joined by word or, 87–88 in one variable, 82–89 Compound interest, 698–699 Concentration rate, 60 Conditional equations, 50, 51 Conic sections See also Circles; Ellipses; Hyperbolas; Parabolas explanation of, 750 on graphing calculators, 787 illustration of, 751 Conjugates complex, 555, 568 explanation of, 341 Consecutive integers, 57 Consistent systems of linear equations in two variables, 241, 305 Constant functions explanation of, 210, 232 identification of, 214 Constant of variation, 470 Constant terms, 30, 39, 165 Continuously compounded interest, 698 Contradictions, 50–51, 111 Cost applications, 264 Cramer’s rule explanation of, A–11, A–12 to solve ϫ systems of linear equations, A–11 – A–12 to solve ϫ systems of linear equations, A–13 – A–14 Cube roots, 494 Cubes perfect, 386 sum and difference of, 382–384, 410 Cubic functions, 213 Curie, Marie, 669 D E Decimals, clearing, 50 Degree of polynomial, 329, 406 Denominators least common, 436–437 rational expressions with like, 435 rational expressions with unlike, 435, 438–441 rationalizing, 534–538, 566 Dependent systems of linear equations analysis of, A–14 in three variables, 291 in two variables, 241, 252–253, 259–260, 291, 301, 305 Determinants explanation of, A–7 on graphing calculators, A–10 of ϫ matrix, A–7 – A–8 of ϫ matrix, A–8 – A–10 Difference of cubes, 382–384, 410 Difference of squares explanation of, 341, 379, 384, 409 to factor binomials, 379–381, 409 in grouping, 381–382 Directrix, of parabola, 751 Direct variation applications involving, 472, 473 explanation of, 470, 485 Discriminant explanation of, 590, 643 graphical interpretation of, 591 method to find, 592, 593 use of, 590–591 Distance with absolute value, 105 rate, time applications, 61–62, 463, 464 between two points, 740, 741, 788 Distance formula, 740–741, 745, 788 Distributive property of multiplication over addition application of, 32–33, 519–521, 525 explanation of, 31, 39 Division of complex numbers, 558 of functions, 223–224 long, 349–352 of polynomials, 349–355, 407–408 of rational expressions, 432, 481 of real numbers, 20–21, 38 synthetic, 352–355 Division algorithm, 351 Division property of equality, 45 of inequality, 75 of radical expressions, 533–534, 566 of radicals, 533–534 Domain of functions, 203–204 of inverse functions, 660–661 of logarithmic functions, 678, 680, 694 of radical functions, 498–499 of rational functions, 426, 480 of relations, 190–193, 230 Dot Mode (graphing calculator), 427 e (irrational number), 7, 8, 37, 493, 697–698, 730 Elementary row operations, 297 Ellipses explanation of, 750, 760, 790 focus of, 760 graphs of, 761–762 standard form of equation of, 760–761 Endpoints, 10, 11 Equality of absolute value, 97 addition property of, 45 division property of, 45 multiplication property of, 45 subtraction property of, 45 Equations See also Linear equations in one variable; Linear equations in three variables; Linear equations in two variables; Quadratic equations; specific types of equations absolute value, 93–97, 116 of circle, 743, 788 conditional, 50 contradictions as, 50–51 equivalent, 44, 75 explanation of, 44 exponential, 710–713, 730 identities as, 51 of inverse of function, 659–661, 727 of lines, 152–160, 180 literal, 68–70, 113, 581–582 logarithmic, 719–723, 731 polynomial, 394–395, 410, 602 in quadratic form, 599–601, 643 radical, 542–548, 567 rational, 451–456, 460–466, 602–603 reducible to quadratic equations, 602–603 solutions to, 44 Equivalence property of exponential expressions, 710 of logarithmic expressions, 719 Equivalent equations, 44, 75 Equivalent rational expressions, 437–438 Exponential equations applications of, 714–715 explanation of, 710, 730 method to solve, 710–713, 730 Exponential expressions equivalence of, 710 evaluation of, 22 explanation of, 21–22 on graphing calculators, 22, 666, 667 Exponential functions applications of, 669–671 decay, 668–670 explanation of, 665–666, 728 graphs of, 667–668, 681 growth, 668, 671 Exponents explanation of, 22, 38 integer, 320, 406 properties of, 320–322 rational, 504–508, 564 simplifying expressions with, 320–323 Subject Index Expressions See Algebraic expressions; Exponential expressions; Logarithmic expressions; Radical expressions; Rational expressions Extraneous solutions, 542 F Factorial notation, A–2 – A–3 Factors/factoring binomials, 361, 379–386, 409–410 difference of squares, 379–381 explanation of, 30 greatest common, 359–364, 369, 408 by grouping, 361–364, 408 negative, 360–361 to solve quadratic equations, 392–395, 410 steps in, 389 sum and difference of cubes, 382–384 within terms, 30 trinomials, 367–375, 408–409 using substitution, 376 Feasible regions, 277–279 Fibonacci, Leonardo, A–26 Fibonacci sequence, A–26 Finite sequences, A–18 First-degree polynomial equations, 391 See also Linear equations in one variable Focus of ellipse, 760 of hyperbola, 762 of parabola, 751 Form fitting, 69 Formulas See also specific formulas explanation of, 25 involving rational equations, 456 as literal equations, 68 Fractions clearing, 48, 49, 452, 453 complex, 444–448, 482 to solve rational equations, 452 Functions See also Inverse functions; Logarithmic functions; Polynomial functions; Quadratic functions; Radical functions; Rational functions absolute value, 213 algebra of, 223–224, 233 combination of, 225 composition of, 224–225, 233 constant, 210, 214, 232 cubic, 213 decreasing, 668 domain of, 203–204, 426, 480, 498–499, 660–661, 678, 680, 694 evaluation of, 200–203 explanation of, 198–199, 231 exponential, 665–671, 728 graphs of, 203, 210–216, 226, 232 identity, 213 increasing, 668 inverse, 656–661, 727 linear, 210, 214, 232 logarithmic, 675–683, 728 maximum value of, 612 minimum value of, 612 nonlinear, 212 notation for, 200–201, 211, 231 one-to-one, 656–658, 727 operations on, 225–226 polynomial, 332–334 quadratic, 213–214, 232, 398, 410, 589 radical, 498–500, 563 rational, 425–427, 480 reciprocal, 213 relations vs., 198–199 square root, 213 vertical line test for, 199–200, 231 x- and y-intercepts of, 215–216 Fundamental principle of rational expressions, 421, 422 Future value, 479 G Gauss-Jordan method explanation of, 297 to solve system of linear equations, 297–300 GCF See Greatest common factor (GCF) Geometric sequences explanation of, A–30 nth term of, A–31 Geometric series applications of, A–34 explanation of, A–32 sum of, A–32 – A–34 Geometry, applications involving, 66–68, 113, 268, 547–548 Graph feature (graphing calculator), 133 Graphing calculator features Dot Mode, 427 Graph, 133 Intersect, 244–245 Maximum, 613 Minimum, 613 Table, 133, 202 Value, 155, 202 Zero, 399 Graphing calculators absolute value function on, 17 change-of-base formula on, 702 circles on, 742 common logarithms on, 678 conic sections on, 787 determinants on, A–10 exponential expressions on, 22, 666–667, 698 factorial notation on, A–3 function values on, 613 linear equations on, 177 logarithms on, 681, 700, 702 matrices on, 301 nonlinear functions on, 212 nonlinear systems of equations on, 773 order of operations on, 25 parabolas on, 754 perpendicular lines on, 159 quadratic inequalities on, 631 radical equations on, 546 radical expressions on, 513 radical functions on, 499 rational exponents on, 505 I-6 scientific notation on, 324 sequences on, A–19, A–22 square roots on, 493, 495 vertical shifts of function on, 607 x- and y-intercepts of functions on, 216 Graphs/graphing of circles, 741–743 of ellipse, 761–762 of exponential functions, 667–668, 681 of feasible region, 277–279 of functions, 203, 210–216, 226, 232 of function values, 203 of hyperbolas, 762–764 of linear equations in two variables, 126–128 of linear inequalities in two variables, 273–275, 308 of logarithmic functions, 678–681 of nonlinear inequalities in two variables, 778–780 of nonlinear systems of equations, 771–774 of parabolas, 751–752, 755 of quadratic functions, 398, 607–613, 644 of radical functions, 499–500 of rational functions, 426 of slope-intercept form of linear equations, 153 of systems of linear equations in two variables, 242–244, 305 of systems of linear inequalities in two variables, 275–279 of systems of nonlinear inequalities in two variables, 780–781 Greatest common factor (GCF) explanation of, 360, 408 factoring out, 359–364, 369 Grouping factoring by, 361–364, 408 use of difference of squares in, 381–382 Grouping symbols, 10, 11, 23 H Half-life, 669, 670, 715 Higher-degree polynomial equations, 394–395 Horizontal asymptotes, 212 Horizontal lines, 131, 132, 178, 180 Horizontal line test, 656–657, 727 Horizontal transverse axis, 763, 790 Hyperbolas explanation of, 750, 790 focus of, 762 graphs of, 762–764 standard forms of equation of, 762–763 I Identities, 51, 111 Identity functions, 213 Identity property of addition, 31 Identity property of multiplication, 31 Imaginary numbers (i) explanation of, 552–553, 568 powers of, 554–555 I-7 Subject Index Inconsistent systems of linear equations Gauss-Jordan method to solve, 300 in three variables, 292 in two variables, 241, 252, 260, 305 Independent systems of linear equations, in two variables, 241, 305 Index, of radical, 494, 528–529 Index of summation, A–22 Inequalities See also specific types of inequalities absolute value, 99–104, 116–117 addition property of, 74 applications of, 78–79 boundary points of, 630–633 compound, 82–89, 115 division property of, 75 explanation of, 8–9 of form a Ͻ x Ͻ b, 86–87, 115 linear in one variable, 74–79, 114 linear in two variables, 273–279, 308 polynomial, 629–633 properties of, 74, 114 quadratic, 629–633 rational, 633–636 with “special case” solution sets, 636–637 strict, 630 subtraction property of, 74 translations involving, 12 Inequality symbols, 9, 75, 274 Infinite sequences, A–18 Integer exponents, 320, 406 Integers consecutive, 57 explanation of, 37 set of, Interest applications involving, 59–60, 266 compound, 698–699 continuous compounding, 698 simple, 58 Intersect feature (graphing calculator), 244–245 Intersection, of sets, 82–84, 115 Interval notation domain in, 204 explanation of, 10–11 Intervals, union and intersection of two, 83–84 Inverse function property, 658 Inverse functions domain and range of, 660–661 explanation of, 656–658, 727 finding equation of, 659–661 graphs of, 661 one-to-one, 656–657 Inverse of functions definition of, 657–658 finding equation of, 659–661, 727 Inverse property of addition, 31 Inverse property of multiplication, 31 Inverse variation applications involving, 473–474 explanation of, 470, 485 Irrational numbers, 7, 8, 37, 493, 697–698, 730 J Joint variation applications involving, 474 explanation of, 471, 485 L LCD See Least common denominator (LCD) Leading coefficient, of polynomial, 329 Leading term, of polynomial, 329, 406 Least common denominator (LCD) multiplication by, 48, 49 of rational expressions, 436–437, 482 Least common multiple, 258 Like terms addition and subtraction of, 32, 33, 406 explanation of, 30, 39 method to combine, 34, 111 Linear equations in one variable applications of, 55–62, 112 clearing fractions and decimals to solve, 48–50 conditional, 50 contradictions as, 50–52 explanation of, 44, 111 identities as, 52 method to solve, 44–48, 111 problem-solving steps for, 55–56, 112 Linear equations in three variables, 286, 309 See also Systems of linear equations in three variables Linear equations in two variables See also Systems of linear equations in two variables applications of, 147, 165–169, 181 explanation of, 125–126, 178 forms of, 159–160, 180 graphs of, 126–128 horizontal and vertical lines and, 131–132 parallel and perpendicular lines and, 145–146 point-slope formula and, 156–159 slope-intercept form and, 152–155, 242, 243 slope of line and, 141–145, 179 solutions to, 126 x- and y-intercepts of, 128–131 Linear functions explanation of, 210, 232 identification of, 214 Linear inequalities in one variable compound, 82–89, 115 explanation of, 74, 114 method to solve, 74–77 Linear inequalities in two variables explanation of, 273, 308 graphs of, 273–275, 308 systems of, 275–279 Linear models interpretation of, 167 key concepts regarding, 181 method to write, 165–166 from observed data points, 168–169 Lines equations of, 152–160, 180 horizontal, 131, 132, 178, 180 in rectangular coordinate system, 241 slope of, 141–145, 179 vertical, 131, 132, 159, 178, 180 x- and y-intercepts of, 129–131 Literal equations See also Quadratic formula application of, 68–69 explanation of, 68, 113 involving rational expressions, 456 method to solve, 69–70, 581–582 Logarithmic equations applications of, 722–723 explanation of, 719, 731 method to solve, 719–722 Logarithmic expressions equivalence property of, 719 evaluation of, 676–677 in expanded form, 691–692, 701 single, 693–694 Logarithmic functions applications of, 682–683 common, 677–678, 728 conversion to exponential form, 675–676 domain of, 678, 680, 694 explanation of, 675–676, 728 graphs of, 678–681 natural, 699–701 Logarithms change-of-base formula and, 702 common, 677–678 explanation of, 675 in exponential form, 675, 676 properties of, 689–691, 729 sum or difference of natural logarithms as single, 693–694, 701 Long division of polynomials, 349–352 synthetic division vs., 355 M Margin of error, 561–562 Mathematical models explanation of, 165, 181 interpretation of, 167 method to write, 165–166 Matrices augmented, 296–298 coefficient, 296 column, 296 determinants of, A–7 – A–10 elementary row operations and, 297 explanation of, 295, 310 order of, 295–296 reduced row echelon form of, 297–298 row, 296 to solve systems of linear equations, 295–297, 310 square, 296 ϫ 2, A–7 – A–8 ϫ 3, A–8 – A–10 Maximum (Graphing calculator feature), 613 Maximum value, of function, 612 Measurement error, 105 Midpoint formula, 744–745 Minimum (graphing calculator feature), 613 Minimum value, of function, 612 Minors of element of matrix, A–8 – A–10 expansion of, A–9 – A–10 Mixture problems, 60–61, 265 Monomials See also Polynomials division by, 349 explanation of, 329, 406 multiplication of, 339–340 Motion problems, 267–268 Subject Index Multiplication associative property of, 31 commutative property of, 31 of complex numbers, 557, 558 distributive property of, over addition, 31, 39 of functions, 223–224 identity property of, 31 inverse property of, 31 of polynomials, 339–344, 407 of radical expressions, 524–529, 565 of rational expressions, 430–431, 481 of real numbers, 19–20, 38 Multiplication property of equality, 45 of inequality, 75 of radicals, 511–514, 524–525, 564, 565 N Natural logarithmic functions applications of, 703 explanation of, 699 graphs of, 699–700 properties of, 700 simplifying expressions with, 700 sum or difference of, 701 Natural numbers, 6, 37 Negative factors, 360–361 Negative numbers, square roots of, 23 Nonlinear functions, 212 Nonlinear inequalities in two variables explanation of, 778, 792 graphs of, 778–780 systems of, 780–781 Nonlinear systems of equations addition method to solve, 773–775 explanation of, 770, 791 on graphing calculators, 773 graphs of, 771–774 substitution method to solve, 770–773 Notation/symbols absolute value bars, 496 factorial, A–2 – A–3 function, 200–201, 211, 231 grouping, 10, 11, 23 inequality, 9, 75, 274 interval, 10–11, 204 radical, 23, 493, 494 scientific, 323–325, 406 set-builder, 6, 10 nth roots approximation of, 495 explanation of, 494–495, 563 on graphing calculators, 495 of nth power, 534–535 of real numbers, 495 of variable expressions, 495–497 nth term of sequence explanation of, A–18, A–27 – A–28, A–31 formula for, A–20 – A–21 on graphing calculators, A–19 Number line See Real number line Numbers See also Integers; Real numbers complex, 555–559, 568 imaginary, 552–555, 568 irrational, 7, 8, 37, 493, 697–698 natural, 6, 37 rational, 7, 37 whole, 6, 37 O One-to-one functions, 656–658, 727 Opposite of polynomial, 330–331 of real number, 16 Ordered pairs in rectangular coordinate system, 125 in relations, 191–192 as solution to linear equation in two variables, 127–128 as solution to systems of linear equations in two variables, 240, 244 Ordered triples, as solution to linear equation in three variables, 286, 287, 309 Order of matrix, 296, 297, 310 Order of operations application of, 24, 25 on calculators, 25 explanation of, 23, 38 simplifying radicals by using, 515 Ordinate See y-coordinate Origin, of rectangular coordinate system, 124 P Parabolas See also Quadratic equations axis of symmetry of, 751 completing the square to to graph, 752–753, 755 directrix of, 751 explanation of, 214, 232, 398, 750, 751, 789 focus of, 751 on graphing calculators, 754 graphs of, 751–753, 755 with horizontal axis of symmetry, 753–754 vertex of, 619–624, 645, 751, 756–757, 789 with vertical axis of symmetry, 751–752 Parallel lines explanation of, 145 in slope-intercept form, 154 slope of, 145, 146 Parentheses clearing, 34 grouping function of, 10, 11, 23 use of, 10, 11 Pascal, Blaise, A–2 Pascal’s triangle, 405, A–2 Percent problems, 58–59 Perfect cubes, 386 Perfect squares, 374, 386, 494 Perfect square trinomials completing the square to create, 578–579 explanation of, 342, 409, 526, 578 factoring of, 374–376 Perimeter application involving, 66–67 formulas for, 113 Perpendicular lines explanation of, 145 on graphing calculators, 159 in slope-intercept form, 154 slope of, 145, 146 I-8 Point-slope formula explanation of, 156, 180 use of, 156–159, 623 Polynomial equations See also Quadratic equations method to solve, 602 zero product rule to solve higher-degree, 394–395, 410 Polynomial functions in applications, 333–334 evaluation of, 333 explanation of, 332 Polynomial inequalities explanation of, 629, 646 graphs to solve, 629–630 test point method to solve, 630–633, 646 Polynomials See also Binomials; Factors/ factoring; Monomials; Trinomials addition of, 330, 406 applications of product of, 343–344 degree of, 329, 406 division of, 349–355, 407–408 explanation of, 329 leading coefficient of, 329 leading term of, 329, 406 multiplication of, 339–344, 407 opposite of, 330–331 prime, 367, 370 subtraction of, 330–332, 406 Population growth, 670, 671 Population model, 726 Power, 22 Power property of logarithms, 690, 691 Powers of i, 554–555 Prime polynomials, 367, 370 Principal and interest problems, 59–60, 266 Principal fourth roots, 544 Principal square roots, 23, 492, 563 Problem solving, 55–56, 112 Product property of logarithms, 689–691 Proportions See also Variation in applications, 461–462 explanation of, 460, 484 methods to solve, 460–461 with similar triangles, 462 Pythagorean theorem applications of, 497–498 explanation of, 497, 563 Q Quadrants, of rectangular coordinate system, 124 Quadratic equations See also Parabolas applications of, 396–397 completing the square to solve, 578–581 discriminant and, 590–593 explanation of, 391, 392 factoring to solve, 392–395, 410 quadratic formula to solve, 586–587, 593 square root property to solve, 576–577, 579–581, 593, 602, 642 summary of methods to solve, 593–595 zero product rule to solve, 392–395, 410, 580, 593 Quadratic form, equations in, 599–601, 643 I-9 Subject Index Quadratic formula applications of, 588–589 derivation of, 585–586 explanation of, 585, 586, 643 to solve quadratic equations, 586–587, 593 to solve rational equations, 602–603 Quadratic functions applications of, 399–400, 589, 623 explanation of, 213–214, 232, 398, 410 of form f(x) ϭ ax2, 610–611 of form f(x) ϭ a(x Ϫ h)2 ϩ k, 611–613, 619–621 on graphing calculator, 607 graphs of, 398, 607–613, 644 (See also Parabolas) x- and y-intercepts of, 591, 592 Quadratic inequalities explanation of, 629 graphs to solve, 629–630 test point method to solve, 630–633, 646 Quadratic models of form f(x) ϭ a(x Ϫ h)2 ϩ k, 641–642 method to write, 624 Quotient property of logarithms, 690, 691 R Radical equations applications of, 547–548 explanation of, 542 on graphing calculators, 546 method to solve, 542–547, 567 Radical expressions addition of, 519–521, 565 division property of, 533–534, 566 on graphing calculators, 513 like, 519, 565 multiplication of, 524–529, 565 multiplication property of, 511–514, 524–525, 564, 565 order of operations to simplify, 515 rationalizing the denominator of, 534–538 simplification of, 496–497, 515, 533–534 simplified form of, 512, 532–533 square of, 526–527 subtraction of, 519–521, 565 Radical functions applications of, 548 domain of, 498–499 explanation of, 498, 563 on graphing calculators, 499 graphs of, 499–500 Radical sign, 23, 493, 494 Radicand, 494, 565 Radius, of circle, 741 Range, of relations, 190–193, 230 Rate of change, 165, 179 Rates, 58–59 Rational equations applications of, 460–466, 484 explanation of, 451 formulas involving, 456 methods to solve, 452–455, 484 Rational exponents applications of, 508 converting between radical notation and, 506 evaluation of, 504–506 explanation of, 504, 505, 564 properties of, 506–507 simplifying expressions with, 507 Rational expressions addition of, 435, 438–441, 482 division of, 432, 481 evaluation of, 420 explanation of, 420, 480 fundamental principle of, 421, 422 least common denominator in, 436–437, 482 multiplication of, 430–431, 481 ratios of -1 and, 424–425 restricted values of, 420–422, 480 simplification of, 421–424, 480, 564 solving literal equations involving, 456 subtraction of, 435, 438–441, 482 writing equivalent, 437–438 Rational functions domain of, 426, 480 evaluation of, 426 explanation of, 425, 480 on graphing calculators, 427 graphs of, 426 Rational inequalities explanation of, 633 graphs of, 637 test point method to solve, 634–636 Rationalizing the denominator explanation of, 534, 566 one term, 534–537, 566 two terms, 537–538, 566 Rational numbers, 7, 37 Ratios of -1, 424–425 common, A–30, A–31 explanation of, 460, 484 Real number line explanation of, intervals on, 10 use of, 8, Real numbers absolute value of, 16, 17 addition of, 17–18, 38 division of, 20–21, 38 explanation of, 6, 37 multiplication of, 19–20, 38 opposite of, 16 ordering, properties of, 30–33 set of, 6–8 subsets of, subtraction of, 18–19, 38 Reciprocal functions, 213 Reciprocals, 20 Rectangles, perimeter of, 66–67 Rectangular coordinate system, 124–125, 241 Reduced row echelon form, 297–298, 310 Relations applications involving, 193–194 domain and range of, 190–193, 230 explanation of, 190, 230 functions vs., 198–199 Restricted values, of rational expressions, 420–422, 480 Right triangles, 397 Roots See also Cube roots; nth roots; Square roots fourth, 544 nth, 494–495 square, 23, 492–494 of variable expressions, 495–497 Row matrices, 296 S Sales tax, 58 Scientific notation application of, 325 explanation of, 323–324, 406 on graphing calculators, 324 writing numbers in, 324 Second-degree polynomial equations, 391 See also Quadratic equations Sequences alternating, A–19 in applications, A–21 arithmetic, A–26 – A–28 explanation of, A–18 Fibonacci, A–26 finite, A–18 geometric, A–30 – A–31 on graphing calculators, A–19, A–22 terms of, A–18 – A–21 Series arithmetic, A–28 – A–30 explanation of, A–22 – A–23 geometric, A–32 – A–34 Set-builder notation, 6, 10 Sets classifying numbers by, explanation of, intersection of, 82–84, 115 in interval notation, 10–11 of real numbers, 6–8 union of, 82–84, 115 Similar triangles, 462 Simple interest, 58 Simplification of algebraic expressions, 33–34, 39 of complex fractions, 444–448 of complex numbers, 559 of exponents, 320–323 of nth roots, 497 of radical expressions, 496–497, 526, 564 of rational exponents, 507 of rational expressions, 421–424, 480 of square roots, 493 in terms of i, 552–553 Slope applications of, 142, 147 explanation of, 141–142 formula for, 142–145 interpretation of, 147 method to find, 142–145 of parallel lines, 145, 146 of perpendicular lines, 145, 146 Slope-intercept form explanation of, 152–153, 180 to find equation of line, 153–155 graph of, 153 linear equations in, 152, 242, 243 Solutions to equations, 44 extraneous, 542 Square matrices, 296 Subject Index Square root functions, 213 Square root property explanation of, 576, 642 to solve quadratic equations, 576–577, 579–581, 593, 602, 642 Square roots evaluation of, 494 explanation of, 23, 492 on graphing calculators, 493, 495 method to simplify, 493 of negative numbers, 23 positive and negative, 492 principal, 23, 492, 563 real-valued, 492 Squares See also Completing the square of binomials, 342, 343, A–1 perfect, 374, 386, 494 sum and difference of, 341, 379–382, 384, 409 Standard form of ellipse equation, 760–761, 790 of hyperbola equation, 762–763, 790 of linear equations, 152, 180 of parabola equation, 753 Straight-line depreciation, 130–131 Study skills, 2–3 Subscripts, 25 Subsets, of real numbers, Substitution method explanation of, 249, 306 to solve equations quadratic in form, 599–601 to solve nonlinear systems of equations, 770–773 to solve systems of linear equations in two variables, 249–251, 306 Subtraction of complex numbers, 556–557 of functions, 223–224 of like terms, 32, 33, 406 of polynomials, 330–332, 406 of radicals, 519–521, 565 of rational expressions, 435, 438–441, 482 of real numbers, 18–19, 38 Subtraction property of equality, 45 Subtraction property of inequality, 74 Sum of arithmetic series, A–30 of geometric series, A–32 – A–34 Summation, index of, A–22 Summation notation, A–22 – A–23 Sum of cubes, 382–384, 410 Sum of squares, 381, 409 Supplementary angles, 113 Symbols See Notation/symbols Synthetic division explanation of, 352–355 long division vs., 355 Systems of linear equations in three variables applications of, 289–291 dependent, 291, 300 explanation of, 286, 309 Gauss-Jordan method to solve, 299–300 inconsistent, 292, 300 matrices to solve, 295–297 methods to solve, 287–289 solutions to, 286–287 Systems of linear equations in two variables addition method to solve, 256–259, 307 applications of, 264–268, 307 consistent, 241, 305 Cramer’s rule to solve, A–11 – A–14 with dependent equations, 252–253, 259–260, A–14 explanation of, 240, 305 graphing to solve, 242–244, 305 inconsistent, 241, 252, 260, 305 with independent equations, 241, 305 solutions to, 240–242 substitution method to solve, 249–251, 306 Systems of linear inequalities in two variables feasible regions and, 277–279 graphs of, 275–277 Systems of nonlinear equations See Nonlinear systems of equations Systems of nonlinear inequalities in two variables, 780–781, 792 T Table feature (graphing calculator), 133, 202 Terms constant, 30 like, 30, 32, 34, 111 of polynomials, 329, 406 of sequence, A–18 – A–21 variable, 30 Test point method for absolute value inequalities, 102–104, 117 for linear inequalities in one variable, 75 for nonlinear inequalities in two variables, 778 for quadratic inequalities, 630–633, 646 for rational inequalities, 634–636 ϫ matrix, A–8 – A–10 Translating to/from English form for absolute value, 105 for compound inequalities, 88–89 for inequalities, 12 for polynomials, 343 for variation, 470–471 Transverse axis, of hyperbola, 762–763 Trial-and-error method explanation of, 369 factoring trinomials by, 369–374, 409 Triangles Pascal’s, 405, A–2 right, 397 similar, 462 Trinomials See also Polynomials AC-method to factor, 367–369, 408 explanation of, 329, 406 leading coefficient of to factor, 373–374 perfect square, 342, 374–376, 409, 526, 578–579 substitution to factor, 376 trial-and-error method to factor, 369–374, 409 ϫ matrix, A–7 – A–8 I-10 U Union, of sets, 82–84, 115 Unlike terms, 30 V Value feature (graphing calculator), 155, 202 Variable expressions, roots of, 495–496 Variables explanation of, 25 isolation of, 46–48 Variable terms, 30, 39, 165 Variation See also Proportions applications of, 472–474 constant of, 470 direct, 470, 472, 473, 485 inverse, 470, 473–474, 485 joint, 471, 474, 485 translations involving, 470–471 Vertex of hyperbola, 762 of parabola, 619–624, 645, 756–757, 789 of quadratic function, 622 Vertex formula, 621, 622, 645, 756–757 Vertical asymptotes, 212 Vertical lines, 131, 132, 159, 178, 180 Vertical line test, 199–200, 231 Vertical transverse axis, 763, 790 W Whole numbers, 6, 37 Word problems, 55–56, 112 See also Applications; Applications Index Work problems, 465, 466 X x-axis, 124 x-coordinate, 125, 128, 131 x-intercepts discriminant and, 591–593 explanation of, 128, 178 of functions, 215–216 of linear equations, 129–131 of quadratic functions, 398, 622 Y y-axis, 124 y-coordinate, 125, 128, 131 y-intercepts explanation of, 128, 178 of functions, 215–216 of linear equations, 129–131, 153 of quadratic functions, 398, 592, 593, 622 Z Zero feature (graphing calculator), 399 Zero product rule explanation of, 392 to solve quadratic equations, 392–395, 410, 580, 593 ... “number sense.” Algebra Intermediate THIRD EDITION N Julie Miller Daytona State College Molly O’Neill Daytona State College Nancy Hyde Broward College— Professor Emeritus INTERMEDIATE ALGEBRA, THIRD... of the copyright page Library of Congress Cataloging-in-Publication Data Miller, Julie, 1962Intermediate algebra / Julie Miller, Molly O’Neill, Nancy Hyde.–Third edition pages cm Includes index... Solving Systems of Linear Equations by the Graphing Method 240 4.2 Solving Systems of Equations by Using the Substitution Method 249 4.3 Solving Systems of Equations by Using the Addition Method 256