This page intentionally left blank Elementary and Intermediate Algebra F o u r t h e d i t i o n Mark Dugopolski Southeastern Louisiana University TM dug84356_fm.indd i 10/28/10 7:01 PM TM ELEMENTARY AND INTERMEDIATE ALGEBRA, FOURTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2012 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2009, 2006, and 2002 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 The McGraw-Hill Companies, Inc., 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 RJE/RJE ISBN 978–0–07–338435–1 MHID 0–07–338435–6 ISBN 978–0–07–735329–2 (Annotated Instructor’s Edition) MHID 0–07–735329–3 Vice President, Editor-in-Chief: Marty Lange Vice President, EDP: Kimberly Meriwether David Senior Director of Development: Kristine Tibbetts Editorial Director: Stewart K Mattson Sponsoring Editor: Mary Ellen Rahn Developmental Editor: Adam Fischer Marketing Manager: Peter A Vanaria Lead Project Manager: Peggy J Selle Senior Buyer: Sandy Ludovissy Senior Media Project Manager: Jodi K Banowetz Designer: Tara McDermott Cover Designer: Greg Nettles/Squarecrow Design Cover Image: © Bryan Mullennix/Alamy Lead Photo Research Coordinator: Carrie K Burger Compositor: Glyph International Typeface: 10.5/12 Times 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 Photo Credits: Page 75: © Vol 141/Corbis; p 82: © Reuters/Corbis; p 150 (top): © George Disario/ Corbis; p 168: © Vol 166/Corbis; p 193 (bottom): © Ann M Job/AP/Wide World Photos; p 246 (bottom left): © Fancy Photography/Veer RF; p 253: © Michael Keller/Corbis; p 255: © DV169/Digital Vision; p 476: © Vol 128/Corbis; p 557: © Daniel Novisedlak/Flickr/Getty RF; p 810: © Vol 168/ Corbis; p 839: © Stockdisc/Digital Vision RF All other photos © PhotoDisc/Getty RF Library of Congress Cataloging-in-Publication Data Dugopolski, Mark Elementary and intermediate algebra / Mark Dugopolski.—4th ed p cm Includes index ISBN 978–0–07–338435–1—ISBN 0–07–338435–6 and ISBN 978–0–07–735329–2—ISBN 0–07–735329–3 (annotated instructor’s edition) (hard copy: alk paper) Algebra—Textbooks I Title QA152.3.D84 2012 512.9—dc22 2010024307 www.mhhe.com dug84356_fm.indd ii 10/28/10 7:01 PM About the Author M ark Dugopolski was born and raised in Menominee, Michigan He received a degree in mathematics education from Michigan State University and then  taught high school mathematics in the Chicago area While teaching high school, he received a master’s degree in mathematics from Northern Illinois University He then entered a doctoral program in mathematics at the University of Illinois in Champaign, where he earned his doctorate in topology in 1977 He was then appointed to the faculty at Southeastern Louisiana University, where he taught for 25 years He is now professor emeritus of mathematics at SLU He is a member of MAA and AMATYC He has written many articles and numerous mathematics textbooks He has a wife and two daughters When he is not working, he enjoys gardening, hiking, bicycling, jogging, tennis, fishing, and motorcycling In loving memory of my parents, Walter and Anne Dugopolski dug84356_fm.indd iii 10/28/10 7:01 PM McGraw-Hill Connect Mathematics McGraw-Hill conducted in-depth research to create a new and improved learning experience that meets the needs of today’s students and instructors The result is a reinvented learning experience rich in information, visually engaging, and easily accessible to both instructors and students McGraw-Hill’s Connect is a Web-based assignment and assessment platform that helps students connect to their coursework and prepares them to succeed in and beyond the course Connect Mathematics enables math instructors to create and share courses and assignments with colleagues and adjuncts with only a few clicks of the mouse All exercises, learning objectives, videos, and activities are directly tied to text-specific material You and your students want a fully integrated online homework and learning management system all in one place McGraw-Hill and Blackboard Inc Partnership ▶ McGraw-Hill has partnered with Blackboard Inc to offer the deepest integration of digital content and tools with Blackboard’s teaching and learning platform ▶ Life simplified Now, all McGraw-Hill content (text, tools, & homework) can be accessed directly from within your Blackboard course All with one sign-on ▶ Deep integration McGraw-Hill’s contentt and content engines are seamlessly woven within your Blackboard course ▶ No more manual synching! Connect assignments ignments within Blackboard automatically (and instantly) feed grades directly to your Blackboard grade center No more keeping track of two gradebooks! Your students want an assignment page that is easy to use and includes lots of extra resources for help Efficient Assignment Navigation ▶ Students have access to immediate feedback and help while working through assignments ▶ Students can view detailed step-by-step solutions for each exercise dug84356_fm.indd iv 10/28/10 7:01 PM Connect Learn Succeed Your students want an interactive eBook rich with integrated functionality Integrated Media-Rich eBook ▶ A Web-optimized eBook is seamlessly integrated within ConnectPlus Mathematics for ease of use ▶ Students can access videos, images, and other media in context within each chapter or subject area to enhance their learning experience ▶ Students can highlight, take notes, or even access shared instructor highlights/notes to learn the course material ▶ The integrated eBook provides students with a cost-saving alternative to traditional textbooks You want a more intuitive and efficient assignment creation process to accommodate your busy schedule Assignment Creation Process ▶ Instructors can select textbook-specific questions organized by chapter, section, and objective ▶ Drag-and-drop functionality makes creating an assignment quick and easy ▶ Instructors can preview their assignments for efficient editing You want a gradebook that is easy to use and provides you with flexible reports to see how your students are performing Flexible Instructor Gradebook ▶ Based on instructor feedback, Connect Mathematics’ straightforward design creates an intuitive, visually pleasing grade management environment ▶ View scored work immediately and track individual or group performance with various assignment and grade reports www.mcgrawhillconnect.com dug84356_fm.indd v 10/28/10 7:01 PM Preface FROM THE AUTHOR I would like to thank the many students and faculty that have used my books over the years You have provided me with excellent feedback that has assisted me in writing a better, more student-focused book in each edition Your comments are always taken seriously, and I have adjusted my focus on each revision to satisfy your needs Understandable Explanations I originally undertook the task of writing my own book for the elementary and intermediate algebra course so I could explain mathematical concepts to students in language they would understand Most books claim to this, but my experience with a variety of texts had proven otherwise What students and faculty will find in my book are short, precise explanations of terms and concepts that are written in understandable language For example, when I introduce the Commutative Property of Addition, I make the concrete analogy that “the price of a hamburger plus a Coke is the same as the price of a Coke plus a hamburger,” a mathematical fact in their daily lives that students can readily grasp Math doesn’t need to remain a mystery to students, and students reading my book will find other analogies like this one that connect abstractions to everyday experiences Detailed Examples Keyed to Exercises My experience as a teacher has taught me two things about examples: they need to be detailed, and they need to help students their homework As a result, users of my book will find abundant examples with every step carefully laid out and explained where necessary so that students can follow along in class if the instructor is demonstrating an example on the board Students will also be able to read them on their own later when they’re ready to the exercise sets I have also included a double cross-referencing system between my examples and exercise sets so that, no matter which one students start with, they’ll see the connection to the other All examples in this edition refer to specific exercises by ending with a phrase such as “Now Exercises 11–18” so that students will have the opportunity for immediate practice of that concept If students work an exercise and find they are stumped on how to finish it, they’ll see that for that group of exercises they’re directed to a specific example to follow as a model Either way, students will find my book’s examples give them the guidance they need to succeed in the course vi dug84356_fm.indd vi 10/28/10 7:02 PM Preface vii Varied Exercises and Applications A third goal of mine in writing this book was to give students more variety in the kinds of exercises they perform than I found in other books Students won’t find an intimidating page of endless drills in my book, but instead will see exercises in manageable groups with specific goals They will also be able to augment their math proficiency using different formats (true/false, written response, multiple-choice) and  different methods (discussion, collaboration, calculators) Not only is there an abundance of skill-building exercises, I have also researched a wide variety of realistic applications using real data so that those “dreaded word problems” will be seen as a useful and practical extension of what students have learned Finally, every chapter ends with critical thinking exercises that go beyond numerical computation and call on students to employ their intuitive problem-solving skills to find the answers to mathematical puzzles in fun and innovative ways With all of these resources to choose from, I am sure that instructors will be comfortable adapting my book to fit their course, and that students will appreciate having a text written for their level and to stimulate their interest Listening to Student and Instructor Concerns McGraw-Hill has given me a wonderful resource for making my textbook more responsive to the immediate concerns of students and faculty In addition to sending my manuscript out for review by instructors at many different colleges, several times a year McGraw-Hill holds symposia and focus groups with math instructors where the emphasis is not on selling products but instead on the publisher listening to the needs of faculty and their students These encounters have provided me with a wealth of ideas on how to improve my chapter organization, make the page layout of my books more readable, and fine-tune exercises in every chapter Consequently, students and faculty will feel comfortable using my book because it incorporates their specific suggestions and anticipates their needs These events have particularly helped me in the shaping of the fourth edition Improvements in the Fourth Edition OVERALL • All Warm-Up exercise sets have been rewritten and now include a combination of fill-in-the-blank and true/false exercises This was done to put a greater emphasis on vocabulary • Using a graphing calculator with this text is still optional However, more Calculator Close-Ups and more graphing calculator required exercises have been included throughout the text for those instructors who prefer to emphasize graphing calculator use • Every chapter now includes a Mid-Chapter Quiz This quiz can be used to assess student progress in the chapter • Numerous applications have been updated and rewritten • All Enriching Your Mathematical Word Power exercise sets have been expanded and rewritten as fill-in-the-blank exercises • All Making Connections exercise sets have been expanded so that they present a more comprehensive cumulative review • Teaching Tips are now included throughout the text, along with many new Helpful Hints dug84356_fm.indd vii 10/28/10 7:02 PM viii Preface CHAPTER • New material on equivalent fractions and reducing fractions • Exercise sets: updated and rewritten applications CHAPTER • Functions are now introduced in the context of formulas • New material on the language of functions • The language of functions and function notation are now used more extensively throughout the text • New material on the simple interest formula, perimeter, and original price applications • New definition of a function and new caution box for the formula and function section • Three updated and rewritten examples to reflect functions in the context of formulas • Exercise sets: 10 updated and rewritten applications • End of chapter: revised and updated summary, review exercises, and chapter test CHAPTER • New material on graphing ordered pairs and ordered pairs as solutions to equations • Simplified introduction to graphing a linear equation in two variables • New material on graphing a line using intercepts • Improved definitions of intercepts and slope intercept form • New material on function notation and applications • Two updated examples and a new caution box • Revised and updated exercise sets for Sections 3.1, 3.3, and 3.4 • Revised and updated Math at Work feature • Exercise sets: updated and rewritten applications • End of chapter: revised and updated review exercises and chapter test • Exercise sets: revised Sections 4.2 and 4.3 to reflect new organization • End of chapter: revised and updated review exercises and chapter test CHAPTER • Section 5.2 has been rewritten with more emphasis on factoring by grouping • Section 5.2 has been reorganized so that factoring by grouping comes before special products • Section 5.5 has been simplified by eliminating division in factoring • New material on factoring applications, factoring by grouping, and the Pythagorean Theorem • New strategy box for factoring a four-term polynomial by grouping • New strategy box for factoring x2 ϩ bx ϩ c by grouping • Three new examples and four revised examples • Rewritten explanation on factoring ax2 ϩ bx ϩ c with a ϭ • New explanation of the sum of two squares prime polynomial • New strategy and explanation for factoring sum and difference of cubes • Revised strategy for factoring polynomials completely • Exercise sets: revised Sections 5.1, 5.5, and 5.6 to reflect new organization • End of chapter: revised and updated review exercises CHAPTER • Updated Section 6.1 by including rational functions • New explanation on rational functions and domain of a rational function CHAPTER CHAPTER • Section 4.2, Negative Exponents and Scientific Notation, has been split into two sections— Section 4.2, Negative Exponents, and Section 4.3, Scientific Notation • Three revised examples and new study tips • New material on using rules for negative exponents • New material on scientific notation, including “Combining Numbers and Words” and “Applications” • New material on polynomial functions dug84356_fm.indd viii • Section 7.1, Solving Systems by Graphing and Substitution, has been split into two sections— Section 7.1, The Graphing Method, and Section 7.2, The Substitution Method • Five updated examples • New summary of the methods for solving systems of equations • Exercise sets: revised Sections 7.3 and 7.4 • End of chapter: revised and updated review exercises and chapter test 10/28/10 7:02 PM dug84356_EOB_ans.qxd A-114 9/22/10 4:19 PM Page A-114 Answers to Selected Exercises R.3 Exercises (4, 0), (0, Ϫ3) (10, 0), (0, Ϫ5) 12 ᎏᎏ 13 Ϫ1 18 19 y y 17 5 1 25 ᎏᎏ, (0, 1) Ϫ4 Ϫ2 Ϫ2 Ϫ3 x Ϫ8 Ϫ4 (3, 0), (0, 6) 10 14 x x Ϫ 2y ϭ 10 2x ϩ y ϭ x Ϫ2 Ϫ3 Ϫ4 (Ϫ3, 0) 3x ϩ 7y ϭ 21 Ϫ2 y Ϫ4 Ϫ2 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ4 Ϫ2 28 2, (0, Ϫ5) yϭ2 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ2 Ϫ3 Ϫ4 Ϫ5 x x x y ϭ 2x Ϫ y Ϫ2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 x 7 x x ϩ 2y ϭ xϪyϭ5 Ϫ2 Ϫ7 Ϫ8 31 ᎏᎏ, (0, Ϫ2) 10 (30, 0), (0, 20) y y y 50 40 5 Ϫ40 Ϫ50 20 10 yϭ x Ϫ 30 x Ϫ20 Ϫ20 Ϫ30 Ϫ40 Ϫ50 20 yϭϪ x 32 ᎏᎏ, (0, 3) y 60 x y ϭ Ϫ4 Ϫ5 20 Ϫ2 Ϫ3 Ϫ4 50 40 30 20 10 Ϫ40 Ϫ20 Ϫ20 xϪ2 30 Ϫᎏᎏ, (0, 2) y Ϫ4 Ϫ2 yϭ 24 Ϫ8 29 1, (0, Ϫ5) Ϫ2 Ϫ3 (60, 0), (0, Ϫ30) Ϫ4 Ϫ2 y ϭ Ϫ3x ϩ 23 x 1 Ϫ4 Ϫ2 xϭ5 y 3 Ϫ2 Ϫ3 Ϫ4 (0, Ϫ4) y y Ϫ4 Ϫ2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 (0, 2) Ϫ4 Ϫ2 x y 1 xϩ1 27 Ϫ3, (0, 4) (5, 0) y Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ2 Ϫ3 Ϫ4 Ϫ5 y y yϭ 16 No slope 21 Ϫᎏᎏ 22 ᎏᎏ 26 ᎏᎏ, (0, Ϫ2) 20 x 15 No slope Ϫ2 Ϫ3 Ϫ4 Ϫ5 y 14 Ϫ4 Ϫ2 (7, 0), (0, 3) y Ϫ4 Ϫ2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 3x Ϫ 4y ϭ 12 Ϫ5 x ϭ Ϫ3 11 60 x ϩ 20 80 x Ϫ4 Ϫ2 3x Ϫ 5y ϭ 10 Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ2x ϩ 3y ϭ x Ϫ7 Ϫ4 Ϫ2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 dug84356_EOB_ans.qxd 9/22/10 4:19 PM Page A-115 Answers to Selected Exercises 34 0, (0, Ϫ5) 33 0, (0, 4) yϭ4 Ϫ4 Ϫ2 5 Ϫ2 Ϫ3 Ϫ4 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 x y ϭ Ϫ5 41 45 48 51 56 57 y y Ϫ4 Ϫ2 Ϫ4 Ϫ2 x Ϫ2 Ϫ3 Ϫ5 Ϫ6 Ϫ7 Ϫ8 4 5 x Ϫ4 Ϫ2 65 66 y y xϽ2 Ϫ4 Ϫ2 Ϫ4 Ϫ2 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 y y x Ն Ϫ1 Ϫ4 Ϫ2 x y 69 x ϩ 3y Յ Ϫ2 Ϫ4 Ϫ2 x Ϫ2 Ϫ2 Ϫ4 Ϫ4 61 6x ϩ y Ն 12 x y y Յ Ϫx ϩ Ϫ4 Ϫ2 y Ն 2x ϩ 1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 x Ϫ4 Ϫ2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 x x 70 y Ϫ2 Ϫ4 Ϫ2 Ϫ4 Ϫ2 yϽ4 Ϫ2 Ϫ3 Ϫ4 Ϫ5 62 y xՅ5 y 1 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ2 Ϫ3 Ϫ4 14 12 10 2 Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ4 Ϫ2 x Ͼ Ϫ3 68 x 60 y Ϫ2 y Ͻ Ϫ2x ϩ Ϫ3 Ϫ4 Ϫ5 67 Ϫ7 Ϫ8 59 y Ͼ 3x Ϫ Ϫ2 Ϫ3 Ϫ4 Ϫ5 x Ϫ y Ͻ Ϫ2 Ϫ3 Ϫ4 Ϫ5 3x Ϫ 2y Ͼ y Ϫ8 36 y ϭ 3x ϩ 37 y ϭ ᎏᎏx ϩ y ϭ Ϫ5x Ϫ 39 y ϭ Ϫ4x ϩ 12 40 y ϭ Ϫᎏᎏx Ϫ 14 yϭ3 42 y ϭ 43 3x Ϫ 5y ϭ Ϫ26 44 14x Ϫ y ϭ 33 x ϩ 2y ϭ 46 x ϩ y ϭ 47 2x Ϫ 3y ϭ 14 x ϩ 4y ϭ Ϫ17 49 x Ϫ 2y ϭ 12 50 x ϩ 4y ϭ Ϫ21 50 mph 52 hr 53 10.5 hr 54 cookies 55 $837.20 $403.20 58 y Ϫ4 Ϫ2 Ϫ6 Ϫ7 35 y ϭ ᎏᎏx Ϫ 38 64 3 Ϫ4 Ϫ2 63 y y A-115 x yϾϪ2 Ϫ4 Ϫ6 Ϫ8 R.4 Exercises Ϫ2x3 ϩ x2 Ϫ 12x w2 ϩ 4w Ϫ x3 ϩ x2 Ϫ 3x 3a2 ϩ 3a Ϫ Ϫ3y2 ϩ y z ϩ 15 Ϫ2t2 ϩ t ϩ 3n2 Ϫ n ϩ 8x2 Ϫ 6x 10 Ϫ30x2 ϩ 10x 11 Ϫ2a3 ϩ 8a2 Ϫ 18a 12 Ϫ6b3 ϩ 15b2 ϩ 3b 13 6w5 Ϫ 6w4 ϩ 6w3 ϩ 18w2 14 10t6 ϩ 5t5 Ϫ 40t4 Ϫ 15t3 15 x2 ϩ 6x ϩ 16 a2 ϩ 12a ϩ 35 17 6s2 Ϫ 7s Ϫ 18 4t2 Ϫ 9t ϩ 19 6x4 ϩ 7x2 Ϫ 20 Ϫ2x6 ϩ 13x4 Ϫ 15x2 x x dug84356_EOB_ans.qxd A-116 21 23 25 27 30 33 36 39 42 45 49 55 58 61 65 71 77 83 89 93 9/22/10 4:19 PM Page A-116 Answers to Selected Exercises x3 Ϫ 18x ϩ 27 22 a3 ϩ 3a2 Ϫ 18a ϩ 16 3w3 ϩ 14w2 ϩ 13w Ϫ 24 Ϫ2m3 Ϫ 10m2 ϩ 19m Ϫ 63 ab ϩ an ϩ mb ϩ mn 26 xy ϩ sx ϩ yt ϩ st x2 Ϫ 4x Ϫ 12 28 x2 Ϫ 2x Ϫ 15 29 6a2 Ϫ 5a Ϫ 15b2 Ϫ 62b ϩ 63 31 10x2 Ϫ 19x ϩ 32 3x2 ϩ 20x ϩ 12 10a6 Ϫ 24a3 Ϫ 18 34 12w6 Ϫ 43w3 ϩ 35 35 16x8 Ϫ x2 25a8 Ϫ x6 37 x2 ϩ 10x ϩ 25 38 y2 ϩ 6y ϩ 4t2 ϩ 28t ϩ 49 40 9w2 ϩ 24w ϩ 16 41 s2 Ϫ 4s ϩ h2 Ϫ 6h ϩ 43 9y2 Ϫ 30y ϩ 25 44 36x2 Ϫ 12x ϩ 9q2 Ϫ 16 46 25m2 Ϫ 36 47 4x4 Ϫ 9n2 48 25t4 Ϫ 9m2 1 2x6 50 4a12 51 Ϫ2w 52 Ϫ4b2 53 ᎏᎏx2 54 Ϫᎏᎏt8 2 3x2 Ϫ 2x ϩ 56 Ϫ5a2 Ϫ 4a Ϫ 57 x3 ϩ 3x2 Ϫ x ϩ 4w2 ϩ 3w Ϫ 59 x2 Ϫ 3x Ϫ 60 2x2 Ϫ 2x ϩ x2 Ϫ 2x Ϫ 62 2x2 Ϫ 3x ϩ 63 x2 Ϫ x ϩ 64 x2 ϩ x Ϫ 3 1 ᎏᎏ 66 ᎏᎏ 67 Ϫᎏᎏ 68 Ϫᎏᎏ 69 Ϫ864a26 70 64b10 12 9a6 x q9 81b4 Ϫ200x12 72 225y16 73 ᎏᎏ6 74 Ϫᎏᎏ6 75 ᎏᎏ 76 ᎏᎏ 8y 8p 16 w6 x4 y12 Ϫ6a6 78 Ϫ15b9 79 ᎏᎏ 80 ᎏ14ᎏ 81 ᎏᎏ 82 ᎏᎏ t 81 25 b8 25 Ϫᎏᎏ 84 Ϫᎏ ᎏ 85 x y 86 ᎏ ᎏ 87 ᎏ ᎏ 88 ᎏᎏ8 32a28 108 a12b20 4x22 y ϫ 1020 90 9.6 ϫ 1025 91 ϫ 101 92 3.6 ϫ 1011 ϫ 1013 94 6.4 ϫ 1022 95 5.12 ϫ 10Ϫ6 96 ϫ 10Ϫ14 R.5 Exercises 4(3x ϩ 2) 6(3a ϩ 5) 3y2(5y Ϫ 2) 16z3(3z Ϫ 2) 4a3b(2b ϩ 5a) 12y3z3(2y ϩ 3z) 4x2(3x2 Ϫ 5x Ϫ 6) 7y(2y2 Ϫ 3y Ϫ 4) 2ab(a2 Ϫ 3a ϩ 3) 10 3wz(w2 Ϫ 4w Ϫ 3) 11 2x 12 5y2 13 x ϩ 14 y2 ϩ 15 Ϫ5a2 16 Ϫ4b2 17 w ϩ 18 y Ϫ 19 (x ϩ 4)2 20 (x ϩ 2)2 21 (a Ϫ 1)2 22 (b Ϫ 5)2 23 (y ϩ 3)(y Ϫ 3) 24 (n ϩ 2)(n Ϫ 2) 25 (3x ϩ 1)2 26 (5y ϩ 2)2 27 (4m Ϫ 5t)2 28 (3s Ϫ 4t)2 29 (3x ϩ 4)(3x Ϫ 4) 30 (9a ϩ 5)(9a Ϫ 5) 31 (8n ϩ 3)2 32 (9s Ϫ 1)2 33 (5x ϩ 7y)(5x Ϫ 7y) 34 (ab ϩ y)(ab Ϫ y) 35 (a ϩ b)(a ϩ 6) 36 (w ϩ x)(w Ϫ 3) 37 (2x ϩ a)(3x Ϫ 5) 38 (5a ϩ 1)(2x ϩ 1) 39 (y2 ϩ 1)(3y Ϫ 4) 40 (3x2 ϩ 5)(2x Ϫ 1) 41 (4a2 ϩ 7)(2a Ϫ 1) 42 (5t2 ϩ 6)(t Ϫ 2) 43 (b Ϫ 3)(a Ϫ 2) 44 (x Ϫ 7)(x Ϫ y) 45 (x2 Ϫ 3)(x Ϫ 1) 46 (x2 Ϫ 5)(a Ϫ 4) 47 (x ϩ 2)(x ϩ 3) 48 (x ϩ 5)(x ϩ 6) 49 (w ϩ 3)(w ϩ 5) 50 (u ϩ 18)(u ϩ 1) 51 (v Ϫ 6)(v ϩ 4) 52 (m Ϫ 11)(m ϩ 2) 53 (t Ϫ 14)(t ϩ 2) 54 (q Ϫ 8)(q ϩ 4) 55 (b Ϫ 13)(b Ϫ 2) 56 ( p Ϫ 25)( p Ϫ 1) 57 (c Ϫ 8)(c Ϫ 3) 58 (n Ϫ 3)(n Ϫ 7) 59 (2x ϩ 3)(x ϩ 2) 60 (3w ϩ 1)(w ϩ 5) 61 (3t ϩ 1)(5t ϩ 4) 62 (6m ϩ 5)((m ϩ 4) 63 (3n Ϫ 2)(n ϩ 6) 64 (4y Ϫ 3)(y ϩ 5) 65 (2m Ϫ 3)(4m ϩ 9) 66 69 72 74 76 78 80 82 84 87 89 91 (3p Ϫ 1)(6p ϩ 5) 67 (4q Ϫ 1)(2q Ϫ 3) 68 (3t Ϫ 4)(2t Ϫ 1) (5z Ϫ 3)(3z Ϫ 2) 70 (k Ϫ 4)(10k Ϫ 1) 71 (x Ϫ 1)(x2 ϩ x ϩ 1) (y Ϫ 3)( y2 ϩ 3y ϩ 9) 73 (a Ϫ 2)(a2 ϩ 2a ϩ 4) (b Ϫ 10)(b2 ϩ 10b ϩ 100) 75 (5x Ϫ 1)(25x2 ϩ 5x ϩ 1) (2a Ϫ 5)(4a2 ϩ 10a ϩ 25) 77 (5q Ϫ 3)(25q2 ϩ 15q ϩ 9) (10b Ϫ 7)(100b2 ϩ 70b ϩ 49) 79 (3x ϩ 4y)(9x2 Ϫ 12xy ϩ 16y2) (2h ϩ 5k)(4h2 Ϫ 10hk ϩ 25k2) 81 (7m ϩ 2n)(49m2 Ϫ 14mn ϩ 4n2) (ab ϩ xy)(a2b2 Ϫ abxy ϩ x2y2) 83 2(x ϩ 1)(x ϩ 3) 3(x ϩ 5)(x Ϫ 3) 85 Ϫ2x(x ϩ 3)2 86 Ϫ4x2(x Ϫ 5)2 3(a Ϫ b)(a ϩ b)(a2 ϩ b2) 88 w(w Ϫ q)(w ϩ q)(w2 ϩ q2) Ϫb(a ϩ 2b)(a2 Ϫ 2ab ϩ 4b2) 90 Ϫ3(2x Ϫ 3)(4x2 ϩ 6x ϩ 9) (a Ϫ 2)(a ϩ 2)(a ϩ 3) 92 (x Ϫ 3)(x ϩ 3)(x Ϫ 5) 93 {Ϫ4, 6} Ά · 1 98 ᎏᎏ, ᎏᎏ 94 {Ϫ5, 4} 99 {Ϫ3, Ϫ2, 0} Ά · 95 Ϫ3, ᎏᎏ Ά · 96 ᎏᎏ, Ά · 97 ᎏᎏ, ᎏᎏ 2 100 {Ϫ1, 0, 2} R.6 Exercises bϪ4 xϩy 2x Ϫ 2a Ϫ 3y5 ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ bϩ4 xϪy xϪ3 aϪ1 2a2 4z7 ᎏᎏ2 Ϫᎏᎏ2 Ϫᎏ3ᎏ2 ᎏᎏ 10 2a ϩ 10 3b 5w 4r t 2(a Ϫ y) x2 ϩ xy ϩ y2 xϩ3 5a 2t3 ᎏ 11 ᎏᎏ 12 ᎏ 13 ᎏᎏ2 14 ᎏᎏ5 xϩy x2 Ϫ 2x ϩ 6b 3w 5b2 3x ϩ 3y 15a4 Ϫ 10a3b 15 ᎏᎏ6 17 ᎏ 18 ᎏᎏ 16 ᎏᎏ5 ᎏ 24a x (x Ϫ y) 3a ϩ 2b 2y xϩ3 ab3 ϩ b4 xy5 ϩ y6 19 ᎏᎏ 20 ᎏᎏ 21 ᎏᎏ 22 ᎏᎏ 23 a ϩ xϪ3 6a4 3(x Ϫ y) Ϫ1 27 28 29 ᎏᎏ 24 w ϩ 25 ᎏᎏ 26 ᎏ2ᎏ x xy xϩy Ϫx Ϫ w2 2t Ϫ t2 30 ᎏᎏ 31 ᎏᎏ 32 ᎏᎏ (x Ϫ 2)(x ϩ 1) (2w ϩ 1)(w Ϫ 3) (3t ϩ 2)(t Ϫ 1) m2 ϩ 8m ϩ n2 ϩ 2n Ϫ 25 33 ᎏᎏ 34 ᎏᎏ 35 ᎏᎏ 36 ᎏᎏ m(m ϩ 1)(m ϩ 3) n(n Ϫ 3)(n ϩ 3) 13 b ϩ 2a Ϫ 15t2 ϩ 30m2 ᎏ 37 ᎏᎏ 38 ᎏᎏ 39 ᎏ 40 ᎏᎏ2 2y Ϫ 5x 8t2 Ϫ 45t m Ϫ 20m 11 20 41 {1} 42 ᎏᎏ 43 {Ϫ9} 44 {2} 45 Ϫᎏᎏ 46 Ϫᎏᎏ 13 11 47 85 teachers 48 cups of cereal 49 48 dogs and 36 cats 50 21 cars and trucks 51 50 mph 52 First day 60 mph and second day 50 mph or first day 50 and second day 40 53 students 54 students Ά · Ά · Ά · dug84356_index.qxd 10/1/10 2:49 PM Page I-1 Index A Absolute value definition of, 532, A:12 equal to negative number, 519–520 equal to positive number, 519–520 equal to zero, 519–520 functions involving, 703–704 of real numbers, 9–10, 27, 28, 77 with roots, 559–560, 573–574 symbolic definition of, 10 Absolute value bars, 40, 44, 45 Absolute value equations, 519–521 absolute value on both sides, 521 summary of, 519, 547 types of, 520 Absolute-value family of functions, 712 Absolute value functions definition of, 702, 771 graphing, 702–703 multiple transformations of, 718 Absolute value inequalities, 521–523, 532–533 applications of, 524 solutions to all real numbers, 523 no real numbers, 524 summary of, 522, 548 in two variables, graphing, 532–533 Absolute value notation, ac method, for factoring trinomials, 347–349, 350, 372, A:47–A:48 Addition associative property of, 59–60, 78, A:16 with polynomials, 282 using, 66–67 commutative property of, 58, 77, A:16 with polynomials, 282 in complex fractions, 419 of complex numbers, 608 distributive property of, A:16 distributive property of multiplication over See Distributive property of multiplication over addition of fractions, 18–21, 76, 407–409, A:13 applications of, 22 vs multiplication, 410 with same denominator, 408 function for, 752, 773 identity property of, 62, 78, A:16 inverse property of, 28, 62, 78, A:16 linear equalities in one variable solved by, 86–88, 90–91, 94–97 nonlinear systems of equations solved by, 841–842 in order of operations, 43, 44 of polynomials, 282, 283, 313 applications of, 284 of radicals, 579–580 of rational expressions, 409–412, 447, A:53–A:54 of rational numbers, 407–409 of real numbers, 26–29, A:13–A:14 applications of, 31 with like sign, 26–27, 77 negative, 26–29 with unlike signs, 27–29, 77 systems of linear equations solved by in three variables, 489, 497 in two variables, 477–487, 482, 497 verbal expressions for, 49, 50, 120 Addition properties of equality, 86–88, 90–91, 94–97, 160, A:20–A:21 simplifying before using, 97–98, 103 of inequality, 152, 153–156, 161 Additive identity, 62, 78 Additive inverse, 28, 62, 78 of polynomials, 290 Algebraic expressions, 49–57, A:15–A:16 applications of, 53, 72 definition of, 49 evaluating, 51 identifying, 49 for pairs of numbers, 121–123, 124 simplifying, 71–72, A:17 translating verbal expressions to, 49–51, 120–129, A:23 involving addition, 49, 50 involving division, 49, 50 involving formulas, 124 involving linear equations, 52–53 involving multiplication, 49, 50 involving subtraction, 49, 50 words used for, 120–121 Algebraic fractions, rewriting, 308–309 Amount formula, 261 “And,” in compound inequalities in one variable, 508, 547 in two variables, 528, 529–530, 547 Angles complementary, 122–123, 131–132 degree measures of, 122–123 sum of, in triangle, 122–123 supplementary, 122–123 Applications of absolute value inequalities, 524 of addition of real numbers, 31 of algebraic expressions, 53, 72 of binomials multiplication, 296 special products, 301 commission problems, 138–139 of complex fractions, 420–421 of compound inequalities in one variable, 513–514 in two variables, 534–535 discount problems, 137–138 of distance, rate, and time, 388, 396–397, A:24–A:25 of equations with exponents, 602–603 of equations with radicals, 602–603 of exponential expressions, 45–46 of exponential functions, 794–796 of factoring out, of greatest common factor, 327 of formulas, 114, 438–439 of fractions, adding, 22 geometric problems, 131–132, A:24 of inequalities, 156 investment problems, 139–140 of linear equations in one variable, 91, 99, 107, 130–144 of linear functions, 177–178 of linear inequalities in two variables, 236–237 of linear programming, 540–541 of logarithmic functions, 806 of logarithms, 824–825 mixture problems, 140–141, A:25 of multiplication property of equality, 91 of nonlinear systems of equations, 843–845 number problems, 130 of order of operations, 45–46 of parabolas, 663 of perimeter, 72, 131 of point-slope form, 215–216 of polynomials addition of, 284 multiplication of, 290–291 subtraction of, 284 of proportions, 433–434, A:55–A:56 of quadratic equations, 653–654 of quadratic formula, 644–645 of quadratic inequalities, 673 of ratio, 430–431, A:55–A:56 of rational expressions, 438–446, A:56–A:57 addition of, 412 rates, 388, 396–397 of scientific notation, 276 of slope, 192–193 of slope-intercept form, 204–205 of special products, 301 of standard form of a line, 204–205 strategies for solving, 130–131, 161 of subtraction of real numbers, 31 of systems of linear equations in three variables, 491–492 in two variables, 462, 470, 482–483 uniform motion problems, 132–133, A:24–A:25 rational expressions in, 388, 396–397 of variation, 226–227 work problems, rational expressions in, 388, 397 Approximately equal to (≈), Approximating irrational numbers, Area in completing the square, 629 geometric models for, 125 of parallelogram, 125 of rectangle, 125 of square, 125 of triangle, 125 Arithmetic, fundamental theorem of, 323 Arithmetic expressions, 40–41 Assets, 26, 27, 29–30, 31 Associative property of addition, 59–60, 78, A:16 with polynomials, 282 using, 66–67 of multiplication, 59–60, 78, A:16 in simplifying, 68–69 I-1 dug84356_index.qxd 10/1/10 2:49 PM Page I-2 I-2 Index Asymptotes of exponential function, 790 of hyperbola, 871 of rational function, 738–741, 773 Axis See x-axis; y-axis Axis of symmetry, of parabola, 853–854 B Balance scale, 86 Base changing, 822–823, 829 definition of, 41 of logarithm, 801 negative, 42–43 Base 10, 789 Base-10 logarithm, 802, 829 Base-a logarithmic function, 801 Base-change formula, 822–823, 829 Base e, 789 Base-e logarithm, 802, 829 Binomial(s) See also Polynomial(s) applications of multiplication, 296 special products, 301 definition of, 280, A:37 as denominator, 410 in equations with rational expressions, 425 expanding, 301 as greatest common factor, 326 higher powers of, 301 multiplication of, 294–298, 314, A:38–A:39 applications of, 296 polynomials divided by, 306–309, 314 as special products, 299–305, 314 square of, 630–631 squaring, 299–300 Boundary line, 232, 234, 235 Bounded intervals, 6–7 Braces, 2, 87, A:2 Brackets in interval notation, and order of operations, 44, 45 Branches, of hyperbola, 871 Building up denominators, 400–402 Building up fractions, 14, 16, 76, 400–401, 402, A:12–A:13 C Calculators See Graphing calculators Cancellation in reducing fractions, 15 in reducing rational expressions, 385 of units, 17 Caret (^), 51 Cartesian coordinate system See Rectangular coordinate system Celsius, 110–111 Center of circle, 861, 888 Center of ellipse, 868 Circle, 861–867, 888 center of, 861, 888 definition of, 861 equation of, 861–863, 888 not in standard form, 863 in standard form, 862–863 graphing, 863 as graph of second-degree inequality, 881 intersection with line, 864 radius of, 861 Closed circle (symbol), Coefficient, 67, 279, A:17, A:37 See also Leading coefficient identifying, 280 leading, 280 Commission applications, 138–139 Common base, 789 Common factor canceling, 15 See also Greatest common factor factoring out, 322–329 opposite of, factoring out, 387–388 Common logarithm, 802, 829 Commutative property of addition, 58, 77, A:16 with polynomials, 282 of multiplication, 58, 77, A:16 in factoring out greatest common factor, 326 using, 66–67 Complementary angles, 122–123, 131–132 Complete factorization, 19, 335–336 in finding least common denominator, 402 of polynomials, 335–336, A:49 in reducing rational expressions, 386–387 strategy for, 357–358, 373 of trinomials, 343, 351–352 Completing the square, 629–633, 643, 678 Complex conjugates, 610 Complex fractions, 417–424, A:54 addition in, 419 applications of, 420–421 definition of, 417, 448 simplifying, 418, 448 strategy for, 418 using least common denominator, 418–420 subtraction in, 419 Complex numbers, 607–615 addition of, 608 definition of, 607–608 division of, 610–611 multiplication of, 608–609 subtraction of, 608 summary of, 613 Composite number, 322 Composition of functions, 753–756, 773 domain in, 754 formulas for, 755 range in, 754 Compound equation, 362 Compound inequalities with “and,” 508 definition of, 145, 508 graphing, 145–146 in one variable, 508–518, 547 with “and,” 547 applications of, 513–514 inequality symbols in, 514 intersection of, 547 with “or,” 547 solution set to, 508–513 solving, 513 union of, 547 with “or,” 508 solving, 155 in two variables, 528–539, 547 with “and,” 528–529, 547 applications of, 534–535 graphing, 529–531, 547 intersection method for, 529, 547 with no solution, 533–534 with “or,” 528–529, 530–531, 533–534, 547 satisfying, 528–529 test-point method for, 530, 531 union method for, 530–531, 547 Compound interest, 114, 794–795 Computations with scientific notation, 275–276 Conditional equations, 106, 160, A:21 Conic sections, 849 See also Circle; Ellipse; Hyperbola; Parabolas Conjugates, 582 complex, 610 multiplication of, 582, 610 rationalizing denominators with, 592 Consecutive even integers, 123–124 Consecutive integers, 123–124 Consecutive odd integers, 123–124 Consistent systems of linear equations in three variables, 490 in two variables, 459–460, 461–462 Constant functions, 692, 701–702, 771 Constant term, 280 Constraints definition of, 540 graphing, 540–541 linear function with maximizing, 541–542, 548 minimizing, 541, 542–543, 548 Continuous-compounding interest, 795–796, 806 Conversion of decimals and fractions to percents, 21 of decimals to fractions, 21 of fractions to decimals, 3, 21 of percents to decimals and fractions, 21 of units of measurement, 17, 110–111 Conversion factors, 17 Coordinate formula for slope, 187–189, 204 Coordinate plane, 170 Coordinates, 3, 170 See also Point(s) Cost joint variation in, 225 linear function for, 177–178 Counting numbers See Natural numbers Cover-up method, 176 Cross-multiplying See Extremes-means property Cube root, definition of, 558 Cubes difference of, factoring, 355–356, 372, A:48–A:49 perfect, 559 sum of, factoring, 355–356, 372, A:48–A:49 Cubic functions, graphing, 725–726 Cubing function, graphing, 725–726 D Debt applications of, 31 expressed as negative number, 2, 26, 27, 28, 29–30 Decimals converting to fractions, 21 converting fractions to, 3, 21 converting to percents, 21 converting percents to, 21 fractions as, 21 in linear equations in one variable, 103–104, A:22 place value for, 21 rational numbers as, repeating, in systems of linear equations in two variables, 481–482 Degree measures of angles, 122–123 Degree of polynomials, 280, 313, A:37 Degree of term, 279 Demand, 462 Demand model, 178 Denominator(s), 13 building up, 400–402 in building up fractions, 14 of equations with rational expressions, variables in, 425 fractions with different, addition and subtraction of, 19–21, 408–409 fractions with same, addition and subtraction of, 18–19, 408 least common See Least common denominator as prime number, 19 rational expressions with different, addition and subtraction of, 410–412 rational expressions with same, addition and subtraction of, 409 rationalizing, 586–587 in reducing fractions, 15 Dependent systems of linear equations in three variables, 490–491 in two variables, 461–462, 497 recognizing, 469 solving by addition, 479–480 solving by graphing, 460 solving by substitution, 469 Dependent variable, in functions, 171–172, 691, 693 dug84356_index.qxd 10/1/10 2:49 PM Page I-3 Index Descartes, René, 170 Difference, 49, 50 See also Subtraction product of, 314 product of sum and, 300, A:39–A:40 square of, 300, 314, A:39–A:40 of two cubes, factoring, 355–356, 372, A:48–A:49 of two fourth powers, factoring, 356–357 of two squares vs a2 ϩ b2, 342 factoring, 332–333, 372 as product of a sum and a difference, 300 Difference function, 752, 773 Directrix of parabola, 851, 853 Direct variation, 223–224, 243, A:33 Discount, 107, 115 Discount applications, 137–138 Discount rate, 115, 137 Discriminant, 644, 650 Distance, rate, and time applications of, A:24–A:25 rational expressions in, 388, 396–397 formula for, 110, 438 variation in, 223, 224 Distance formula, 110, 850, 887 Distributive property of addition, A:16 of multiplication, A:16 Distributive property of multiplication over addition, 60–62, 78 with additive inverses of polynomials, 290 in FOIL method, 294 with higher powers of binomials, 301 with polynomials, 282, 289, 305–306 using, 67 Dividend, 308, A:40 Dividing out in reducing fractions, 15 in reducing rational expressions, 385 Division of complex numbers, 610–611 of exponential expressions, 257–258 of fractions, 17–18, 76, A:13 function for, 752, 773 linear equations in one variable solved by, 89–90 long, 306–307 negative numbers in, 36 of polynomials, 307–308 of monomials, 305 in order of operations, 43, 44 of polynomials, 305–311, 314, A:40 by binomials, 306–309 by monomials, 305–306 of radicals, with same index, 590–592 of rational expressions, 394–396, 447, A:52–A:53 of rational numbers, 394 of real numbers, 35–37, 77, A:14–A:15 with like sign, 35, 36, 37 with unlike signs, 35, 36, 37 verbal expressions for, 49, 50, 121 by zero, 37 of zero, 36, 37 Divisor, 18, 308, 322, A:40 Domain in composition of functions, 754 of exponential functions, 789, 790 of function, 384, 695–696, 701, 771 of inverse functions, 760 of linear functions, 541 of logarithmic functions, 801, 802–803 of radical expression, 563–564, 616 of radical function, 564, 616 of rational expressions, 383–384 of rational function, 738 of relation, 695–696 Downward-opening parabola, 659–660, 852 Downward translation, 716 E e (number), 788–789 Elements, of set, A:2 Elimination in nonlinear systems of equations, 840–843 in systems of linear equations in three variables, 487–490, 497 Ellipse, 868–870, 888–889 center of, 868 definition of, 868 equations of centered at (h,k), 869–870, 889 centered at origin, 869, 888 graphing, 869 sketching, 869 Empty set, 106, A:4 Endpoints of finite intervals, of infinite intervals, of intervals, A:11 Equality addition property of, 86–88, 94–97, 160, A:20–A:21 simplifying before using, 97–98, 103 multiplication property of, 88–90, 94–95, 160, A:20–A:21 applications of, 91 simplifying before using, 97–98, 103 of two ratios See Proportion(s) Equality symbol (ϭ), 52 Equal sets, A:3 Equations, 52–53, 160 absolute value See Absolute value equations as balance scale, 86 of a circle, 861–863, 888 not in standard form, 863 in standard form, 862–863 compound, 362 conditional, 106, 160, A:21 definition of, 52, 86, A:20 direct variation in, 224 of ellipse centered at (h,k), 869–870, 889 centered at origin, 869, 888 equivalent, 86, 160, A:20 with even-root property, 597–598 exponential See Exponential equations with exponents, 596–606, 617 applications of, 602–603 fourth-degree, 651 of hyperbola centered at (h,k), 874–875, 889 centered at origin, opening left and right, 871–872, 889 centered at origin, opening up and down, 872–873, 889 identities, 105, 107, 160 imaginary solutions to, 612–613 inconsistent, 106, 107, 160, A:21 of a line, 438 linear See Linear equations literal See Formulas logarithmic See Logarithmic equations with odd-root property, 596 of parabola, 851–852 changing form of, 854–855 in form x ϭ a(y Ϫ k)2 ϩ h, 855–856, 887 in form x ϭ ay2 ϩ by ϩ c, 888 in form y ϭ a(x Ϫ h)2 ϩ k, 852–855, 887 in form y ϭ ax2 ϩ bx ϩ c, 854–855, 888 process for solving, simplifying, 104 quadratic See Quadratic equations quadratic in form, 650–652, 678 with radicals, 596–606, 617 applications of, 602–603 raising each side to a power, 598–600 with rational exponents, 601–602 with rational expressions extraneous solutions to, 426–427 quadratic equations from, 634 solving, 424–429, 448, A:55 with two solutions, 425–426 variables in denominators of, 425 relations as, 694 solutions to, 52, 86, 87, A:20 squaring both sides of, twice, 600–601 Equilibrium price, 462 Equivalent equations, 86, 160, A:20 Equivalent fractions, 13–15 in adding and subtracting fractions, 19 Equivalent inequalities, 152, 153 Equivalent ratios, 430 Even powers, inverse of functions with, 765 Even-root property, 597–598, 629, 634, 643, 678 Even roots, 558 inverse of functions with, 765 I-3 Exponent(s) caret as symbol of, 51 definition of, 41, 77 equations with, 596–606, 617 applications of, 602–603 fractional See Rational exponents negative, 264–272, 312, A:42 integral, 264–267, 312 rules for, 265, 267–269, 312 in nonlinear systems of equations, 842 in perfect cubes, 559 in perfect fourth powers, 559 in perfect squares, 559 in polynomials, 279 rational See Rational exponents rules for, 256–264, 312, A:41 negative exponents, 265, 267–269, 312, A:42 power of a power rule, 258–259, 269, 312, A:41 power of a product rule, 259–260, 312, A:41 power of a quotient rule, 260, 269, 312, A:41 product rule, 256, 268–269, 312, 313, A:38, A:41 quotient rule, 257–258, 268–269, 312, A:40, A:41 for rational exponents, 572–573, 617 with scientific notation, 275–276 summary of, 261, 268 zero exponent, 257, A:41 zero, 257, A:41 Exponential equations, 792–794 solving, 793–794, 821–822 with different bases, 822 with powers of same base, 821 with single exponential expression, 821 strategy for, 823, 829–830 Exponential expressions, 41–43, A:15 See also Scientific notation applications of, 45–46 definition of, 41 division of, 257–258 evaluating, 42–43 multiplication of, 256 with negative exponents, 266–267 negative numbers in, 42–43 with rational exponents, evaluating, 568 roots of, 560 Exponential functions, 788–800 applications of, 794–796 asymptotes of, 790 bases of, 788–789 between and 1, 790–791 greater than 1, 789–790 definition of, 788, 829 domain of, 789, 790 graphing, 789–791 inverse of, 804, 811 in nonlinear systems of equations, 842 as one-to-one functions, 792–793 one-to-one property of, 793 dug84356_index.qxd 10/1/10 2:49 PM Page I-4 I-4 Index Exponential functions (continued) transformations of, 791–792 x-coordinate of, 794 y-intercept of, 790 Exponential notation, 41 Expressions algebraic See Algebraic expressions arithmetic, 40–41 definition of, A:15 exponential See Exponential expressions rational See Rational expressions simplifying, 66–75 Extraneous solutions, 426–427, 599 Extremes, 431, A:55 Extremes-means property, 432, 448, A:55 F Factor(s), 34 binomial, 326 common See Common factor definition of, 322, A:45 of difference of two fourth powers, 356–357 greatest common See Greatest common factor of quadratic equations, correspondence with solutions, 649–650 Factoring, 321–379, 628, A:45–A:51 See also Factorization in addition and subtraction of fractions, 19 completely See Complete factorization definition of, 322 of difference of two cubes, 355–356, A:48–A:49 of difference of two squares, 332–333, 372, A:45–A:46 of equations containing radicals, 633–634 by grouping, 330–332, 372, A:46–A:47 of large numbers, 323 of perfect square trinomials, 333–335, 372, A:45–A:46 quadratic equations solved by, 361–371, 373, 643, 678, A:49–A:50 applications of, 366–367 strategy for, 361 of rational expressions, in multiplication of, 393 in reducing fractions, 15 of sum of two cubes, 355–356, A:48–A:49 of trinomials ac method for, 347–349, 350, 372, A:47–A:48 ax2 ϩ bx ϩ c with a ϭ 1, 339–346, A:47 ax2 ϩ bx ϩ c with a 1, 347–358, A:47–A:48 perfect square trinomials, 333–335, 372, A:45–A:46 trial and error method for, 349–351, 373, A:48 with two variables, 343 Factoring out common factors, 322–329, A:45 greatest common factor, 325–326, 372 applications of, 327 opposite of common factor, 387–388 opposite of greatest common factor, 326–327, 352 radicals simplified by, 587 Factoring tree, 322 Factorization complete, 19 in finding least common denominator, 402 of polynomials, 335–336, A:49 in reducing rational expressions, 386–387 strategy for, 357–358, 373 of trinomials, 343, 351–352 prime, 322–323 Fahrenheit, 2, 110–111 Family of functions, 712 Finite intervals, 6–7 Finite sets, A:2 Focus (foci) of ellipse, 868 of hyperbola, 870 of parabola, 851, 853 FOIL method, 294–296, 314, A:38–A:39 Formulas, 110–111, A:23 amount, 261 applications of, 114, 438–439 area of parallelogram, 125 area of rectangle, 125 area of square, 125 area of triangle, 125 base-changing, 822–823, 829 for compositions of functions, 755 compound interest, 795 continuous-compounding, 795–796 coordinate formula for slope, 187 definition of, 110, 160 discount, 115 distance, 850, 887 distance, rate, and time, 110, 438 finding value of variable in, 113, 439 functions expressed by, 691–692 geometric, 114–115 interest rate, 114 midpoint, 850–851, 887 perimeter, 114–115 present value, 269–270 quadratic See Quadratic formula resistance, 439 rewriting for one variable, 110–113, 160, A:23 sale price, 115 slope, 187 solving linear equations in one variable with, 110–119 uniform motion, 438, 439–440 using, 124 Four-term polynomials, factoring by grouping, 330–332 Fourth-degree equations, 651 Fourth powers difference of, factoring, 356–357 perfect, 559 Fraction(s), 13–25, A:12–A:13 See also Rational numbers addition of, 18–21, 76, 407–409, A:13 applications of, 22 vs multiplication, 410 with same denominator, 408 algebraic, rewriting, 308–309 building up, 14, 16, 76, A:12–A:13 as coefficients, in linear equations in one variable, 90 complex See Complex fractions converting to decimals, 3, 21 converting decimals to, 21 converting to percents, 21 converting percents to, 21 definition of, 13 division of, 17–18, 76, A:13 equivalent, 13–15 improper, 18 least common denominator of, 19, 77, 102 in linear equations in one variable, 102, A:22 in lowest terms, 15, A:12–A:13 in mixed numbers, 18 multiplication of, 15–16, 76, A:13 vs addition, 410 in quadratic equations, converting to integers, 365–366 ratios as, 430 reducing, 15, 76, A:12–A:13 simplifying to lowest terms, 15 slash as symbol of, 51 subtraction of, 18–21, 76, 407–409, A:13 with same denominator, 408 in systems of linear equations in two variables, 480–481 Fractional exponents See Rational exponents Fraction bars in complex fractions, 417 in division, 37 as grouping symbol, 40, 44 in rational expressions, 395–396 Function(s), 111–113, 771 absolute value See Absolute value functions combining, 751–759, 773 composition of, 753–756, 773 concept of, 690 graphing, 701 constant, 692, 701–702, 771 definition of, 111, 161, 771 determining, 690–691 difference, 752 domain of, 384, 695–696, 701, 771 exponential See Exponential functions expression of by formulas, 691–692 by ordered pairs, 693–694 by tables, 692–693 families of, 712 graphing, 701–711 identity, 701 input of, 690 inverse See Inverse functions invertible, 760 involving absolute value, 703–704 linear See Linear function logarithmic See Logarithmic functions as model, 691 one-to-one, 761, 773 operations with, 751–753 output of, 690 piecewise, 706 polynomial See Polynomial functions product, 752 quadratic See Quadratic functions quotient, 752 range of, 695–696, 701, 771 rational See Rational functions as relation, 693 as rule, 691 square-root, 705, 771 sum, 752–753 transformation of, 712–723, 772 horizontal translation, 712–713, 772 multiple, 716–718 reflection, 714–715, 772 stretching and shrinking, 713–714, 772 translation, 772 vertical translation, 715–716, 772 vertical-line test for, 694–695, 771 Function notation, 177, 696–697, 771, 774 polynomials in, 281–282 Fundamental rectangle of hyperbola, 871 Fundamental theorem of algebra, 651 Fundamental theorem of arithmetic, 323 G Gauss, Carl Friedrich, 651 GCF See Greatest common factor Geometric applications, 131–132, A:24 Geometric formulas, 114–115 Geometric models for area, 125 for perimeter, 125 Geometry See also Area; Perimeter function in, 692 review of, A:1 Golden rectangle, 627, 657 Graph(s) of intervals, of intervals of real numbers, of linear equations in one variable, 173 Graphical method for polynomial inequalities, 730–731, 772 for quadratic inequalities, 668–671, 679 for rational inequalities, 743, 773 dug84356_index.qxd 10/1/10 2:49 PM Page I-5 Index Graphing absolute value functions, 702 absolute value inequalities, in two variables, 532–533 circle, 863 compound inequalities, 145–146 in one variable, 509–513 in two variables, 529–531, 547 constant functions, 701 constraints, 540–541 cubic functions, 725–726 ellipse, 869 exponential functions, 789–791 functions, 701–711 See also Graphing, polynomial functions absolute value, 702 constant, 701 cubic, 725–726 exponential, 789–791 horizontal translation, 712–713 inverse, 765–766, 774 linear, 701–702 logarithmic, 803–804 multiple transformations, 716–718 piecewise, 706 polynomial, 725–738, 772 quadratic, 658–668, 679 rational, 738–751 reflection of, 714–715 square-root, 705 stretching and shrinking, 713–714 transformation of, 712–723, 772 vertical translation, 715–716 horizontal lines, 175 hyperbola centered at (h,k), 874–875 centered at origin, opening left and right, 871–872 centered at origin, opening up and down, 873–874 inequalities, 145 See also Graphing, linear inequalities absolute value, 532–533 compound, 145–146, 509–513, 529–531, 547 second-degree, 881–882 integers, inverse functions, 765–766, 774 linear equations in two variables, 172–176 with intercepts, 176–177, A:29 from point and slope, 189–190, 243 with slope-intercept form, 201–203 summary of, 237 linear functions, 701–702 linear inequalities in two variables, 232–234, 243, A:33–A:34 applications of, 236–237 strategy for, 233 summary of, 237 logarithmic functions, 803–804 nonlinear systems of equations, 840 ordered pairs, 170–171 parabolas, right-opening, 856 parallel lines, 190–191 perpendicular lines, 191–192 piecewise functions, 706 points, 170–171 polynomial functions, 725–738 behavior at x-intercepts, 728–729, 772 cubic functions, 725–726 quartic functions, 726–727 symmetry in, 727–728, 772 transformations of, 730 quadratic functions, 658–668, 679 rational functions, 738–751 sketching, 741–742 relations, 706–707 second-degree inequalities, 881–882, 889 square-root functions, 705 systems of linear equations in two variables, 458–461, 482, 497 systems of second-degree inequalities, 890 vertical lines, 175 Graphing calculators absolute value equations on, 520 absolute value functions on, 703 absolute value inequalities on, 522, 523 addition of radicals on, 580 addition of signed numbers on, 29 base 10 on, 789 base-10 logarithm on, 803 base-changing on, 822 base e on, 789 base-e logarithm on, 803 checking inequalities on, 146 checking solutions to equations on, 98 circles on, 863 common logarithm on, 802 completing the square on, 631, 632 complex fractions on, 419 complex numbers on, 609 composition of functions on, 754 compound inequalities in one variable on, 512, 513 compound interest on, 795 decimals on, converting to fractions, 21 division of fractions on, 18 ellipse on, 869 equations with radicals on, 599, 600 equations with rational exponents on, 601 evaluating polynomials on, 282 exponential equations on, 793 exponential expressions on, 42 exponential functions on, 789, 790, 791 fraction feature on, 15, 16 fractions on adding, 21 addition of, 21 converting to decimals, 21 division of, 18 multiplication of, 16 reducing, 15 subtraction of, 21 function notation on, 697 grouping symbols within grouping symbols on, 45 horizontal translation on, 712–713 hyperbola on, 872 irrational numbers on, approximating, linear equations in two variables on with intercepts, 177, 189–190 with ordered pairs, 174 with slope-intercept form, 202 table of values for, 173 logarithmic equations on, 820 logarithmic functions on, 803 logarithms on, 802 multiple transformations on, 717 multiplication of fractions on, 16 multiplication of radicals on, 583 multiplication of real numbers on, 35 multiplicative inverses on, 63 natural logarithm on, 802 negative exponents on, 265, 267 negative rational exponents on, 571 negative sign on, 29, 70 nonlinear systems of equations on, 843 order of operations on, 44, 45 parabolas on, 659, 663, 854–855 parabola vertex on, 855 parentheses on, 41, 70 perpendicular lines on, 204, 215 point-slope form on, 213 polynomial functions on behavior at x-intercepts, 728–729 symmetry of, 728 transformations of, 730 power key on, 42 power rule for logarithms on, 813–814 product rule for logarithms on, 812 product rule for radicals on, 560 properties of logarithms on, 815 quadratic equations on, 634, 635, 649, 652 quadratic formula on, 641, 642, 643 quadratic functions on, 704 quadratic within a quadratic on, 652 quotient rule for logarithms on, 813 quotient rule for radicals on, 563 radical expressions on, 587 radicals added on, 580 radicals multiplied on, 583 raising each side to a power on, 599, 600 rational exponents on, 568 rational expressions on, 382 rational function asymptotes on, 739 rational functions on, 742 reducing fractions on, 15 reflection on, 714–715 roots on, 559 scientific notation on, 273, 274, 276 shrinking on, 713, 714 standard viewing window on, 189 I-5 stretching on, 713, 714 systems of linear equations in three variables on, 488, 489, 492 systems of linear equations in two variables on, 458, 459, 462, 467, 469, 470, 477, 478, 480, 481 TABLE feature on, 153 TEST menu on, 144 verifying inequalities on, 144, 153 vertical lines on, 175 viewing window of, 204 Graph paper, 171 Greater than (Ͼ), 144 Greater than or equal to (у), 144 Greatest common factor (GCF), 323–324, A:45 binomial as, 326 canceling, 385 dividing out, 385 factoring out, 325–326, 352, 372 applications of, 327 by grouping, 330–332 vs least common denominator, 404 for monomials, 324–325, 372 opposite of, factoring out, 326–327 in reducing rational expressions, 385 strategy for finding, 323 Grouping, factoring by, 330–332, 372, A:46–A:47 of ax2 ϩ bx ϩ c with a ϭ 1, 339 Grouping symbols, 40–41, 43, 44–45 H Horizontal asymptote, of rational function, 738–741 Horizontal boundary lines, 234 Horizontal lines graphing, 175 slope of, 188, 242 Horizontal-line test, 761–762, 774 accuracy of, 762 Horizontal translation, 712–713, 772 Hyperbola, 870–875, 889 asymptotes of, 871 branches of, 871 definition of, 870 equations of centered at (h,k), 874–875, 889 centered at origin, opening left and right, 871–872, 889 centered at origin, opening up and down, 872–873, 889 foci of, 870 fundamental rectangle of, 871 graphing, 871–874, 875 as graph of second-degree inequality, 882 Hypotenuse, 367 I Identity, 105, 107, 160, 282, A:21 Identity function, 701 dug84356_index.qxd 10/1/10 2:49 PM Page I-6 I-6 Index Identity property of addition, 62, 78, A:16 of multiplication, 62, 78, A:16 Imaginary numbers, 597 definition of, 607 powers of, 609 as quadratic equation solutions, 634–635 Imaginary part of complex numbers, 607 Imaginary solutions, 612–613 Improper fractions, 18 Inconsistent equations, 106, 107, 160, A:21 Inconsistent systems of linear equations in three variables, 490–491 in two variables, 461–462, 497 recognizing, 469 solving by addition, 479–480 solving by graphing, 460–461 solving by substitution, 469 Independent systems of linear equations in three variables, 490 in two variables, 461–462, 497 solving by addition method, 477 solving by graphing, 459–460 Independent variable, in functions, 171–172, 691, 693 Index of the radical, 558 Inequalities, 144–151, A:26 absolute value See Absolute value inequalities addition property of, 152, 153–156, 161 applications of, 156 checking, 146 compound See Compound inequalities equivalent, 152, 153 graphing, 145 linear See Linear inequalities multiplication property of, 152, 153–156, 161 on number line, 144, 145 polynomial See Polynomial inequalities quadratic See Quadratic inequalities rational See Rational inequalities rules for, 152–153 second-degree, 881–886, 889–890 definition of, 881 graphing, 881–882 solution sets to, 882, 889 systems of, 882, 890 as seesaw, 152 simple, 508 solving, 153–156 verifying, 144 writing, 147 Inequality symbols, 144, A:26 on compound inequalities, 155–156 in linear inequalities in two variables, 231, 232, 233 reversing, 154 using two, 514 Infinite (unbounded) interval, 7–8 Infinite intervals, 7–8 Infinite sets, A:2 Infinity (ϱ), 7, 145 Infinity symbol (ϱ), 7–8, A:11 Integers, 2, A:11 consecutive, 123–124 graphing, in rational numbers, 2–3 ratio of, 429 set of, 2, 76 variables as, Integral exponents See Exponent(s) Intercepts See x-intercept; y-intercept Interest compound, 114, 794–795 continuous compounding, 795–796 simple, 114 Interest rate, 114 Intersection method, 529 Intersection of sets (ʝ) for compound inequalities in one variable, 509, 512, 547 for compound inequalities in two variables, 529–530, 547 definition of, A:4 Interval notation, 6–8, 76, A:11–A:12 Intervals endpoints of, 6, A:11 finite, graphs of, 6, infinite, 7–8 overlapping, 510–511 of real numbers, 6–8, 76, A:11 unbounded, 7–8 Inverse functions, 760–770, 773–774, 829 definition of, 761 domain of, 760 function notation for, 771, 774 for function with even powers, 765 for function with even roots, 765 graphing, 765–766, 774 horizontal-line test for, 761–762, 774 identifying, 761, 762–763 range of, 760 switch-and-solve strategy for, 763–765, 774 Inverse properties of addition, 28, 62, 78, A:16 of logarithms, 811–812, 829 of multiplication, 62–63, 78, A:16 Inverse variation, 224–225, 243, A:33 Investment amount formula for, 261 applications of, 139–140 present value formula for, 269–270 Irrational numbers approximating, definition of, 5, 76, A:11 as denominator, rationalizing, 586–587 as exponents, in exponential functions, 789 as quadratic equation solution, 632–633 symbols for, Irreducible polynomials See Prime polynomials Isolated variables, 86 J Joint variation, 225, 243, A:33 L Last terms, 629–630 Latitude, 170 Leading coefficient, 280 Least common denominator (LCD) of complex fractions, simplifying with, 418–420 definition of, 402 finding, 19 of fractions, 19, 77 in linear equations in one variable, 102 vs greatest common factor, 404 of polynomials, strategy for finding, 403 of rational expressions, 400–407 converting to, 403–404 finding, 402–403, 447 in solving equations with rational expressions, 425 in solving rational inequalities, 743 Left, translation to, 712 Left-opening parabola, 855–856 Less than (Ͻ), 144 Less than or equal to (р), 144 Like radicals, 579 Like terms combining, 67–68 definition of, 67, A:17 in linear equations in one variable, 97–98 Line(s) boundary, 232, 234, 235 equations of See Linear equations in one variable; Linear equations in two variables as graph of linear equation in two variables, 172 horizontal See Horizontal lines intersection with circle, 864 number See Number line parallel See Parallel lines perpendicular See Perpendicular lines point-slope form of See Point-slope form regression, 184 slope-intercept form of See Slopeintercept form slope of See Slope of line vertical See Vertical lines Linear equations, systems of See Systems of linear equations Linear equations in one variable, 85–168, A:20–A:28 applications of, 91, 99, 107, 130–144 conditional, 106, 160, A:21 with decimals, 103–104, A:22 definition of, 87, 160, A:20 of form ax ϩ b ϭ 0, 94–95 of form ax ϩ b ϭ cx ϩ d, 95–97 with fractional coefficients, 90 with fractions, 102, A:22 graph of, 173 identity, 105, 107, 160, A:21 inconsistent, 106, 107, 160, A:21 with parentheses, 97–98 solving, 94–101, 160, A:20–A:22 with addition property of equality, 86–88, 90–91, 94–97 by division, 89–90 of form ax ϩ b ϭ 0, 94–95 of form ax ϩ b ϭ cx ϩ d, 95–97 with multiplication property of equality, 88–90, 94–95 order of operations for, 95 simplifying, 97–98 simplifying process for, 104 strategy for, 98, 160 by subtraction, 87 translation of verbal expressions for, 52–53, 120–129 with variables on both sides, 90–91 writing, 124–125 Linear equations in two variables, 169–254 See also Variation definition of, 172, A:29 form of, 173–174 graphing, 172–176 with intercepts, 176–177, A:29 from point and slope, 189–190, 243 with slope-intercept form, 201–203 summary of, 237 in nonlinear system of equations, 840 ordered pairs as solutions to, 171–172 of parallel lines, 213–214 of perpendicular lines, 214–215 point-slope form of See Point-slope form slope-intercept form of See Slopeintercept form solution set to, 172 standard form of, 200–201, 203, 242, A:30–A:31 systems of See Systems of linear equations, in two variables writing in point-slope form, 212, 242 in slope-intercept form, 203–204, 242 through two points, 213 Linear function, 701–702, 771 applications of, 177–178 with constraints maximizing, 541–542 minimizing, 541, 542–543, 548 definition of, 177, 243, 692, 771 domain of, 541 graphing, 701–702 in linear programming, 541 of two variables, 541 Linear inequalities in one variable, test-point method for, 236 dug84356_index.qxd 10/1/10 2:49 PM Page I-7 Index Linear inequalities in two variables, 231–241, 243 definition of, 231, A:33 graphing, 232–234, 243, A:33–A:34 applications of, 236–237 strategy for, 233 summary of, 237 in linear programming See Constraints satisfying, 231–232 solution set to, 232 test-point method for, 235–236, 243 Linear programming, 540–546, 548 applications of, 540–541 constraints of, graphing, 540–541 linear functions in, 548 maximizing in, 541–542, 548 minimizing in, 541, 542–543, 548 principle of, 541 strategy for, 542 Line segment length of, 850–851, 887 midpoint of, 850–851, 887 Literal equations See Formulas Logarithm(s) applications of, 824–825 changing base of, 822–823 combining, 816 common, 802, 829 definition of, 800 finding, 802 natural, 802, 829 in nonlinear systems of equations, 842–843 one-to-one property of, 804–805 properties of, 811–819, 829 inverse properties, 811–812, 829 power rule, 813–814, 829 product rule, 812, 829 quotient rule, 813, 829 using, 814–816 Logarithmic equations, 804–805 solving, 819–821 with one logarithm, 819 strategy for, 823, 829–830 using one-to-one property, 820–821 using product rule, 819–820 Logarithmic functions, 800–809 applications of, 806 base-a, 801 base between and 1, 804 definition of, 801, 829 domain of, 801, 802–803 graphing, 803–804 inverse of, 804 in nonlinear systems of equations, 842–843 as one-to-one function, 804–805 range of, 801, 802–803 Long division, 306–307 negative numbers in, 36 of polynomials, 307–308, A:40 Longitude, 170 Lowest terms fractions in, 15, A:12–A:13 rational expressions in, 384–386, A:52 M Maximizing, 541–542, 548 Maximum value, of parabola, 660 Means (in proportion), 431, A:55 Members, of set, A:2 Memory devices, 43 Midpoint formula, 850–851, 887 Minimizing, 541, 542–543, 548 Minimum value, of parabola, 660 Mixed numbers, 18 Mixture problems, 140–141, A:25 Mnemonics, 43 Model, 53, 124, 691 Monomials See also Cubes; Polynomial(s) definition of, 280, A:37 dividing polynomials by, 305–306 division of, 305 greatest common factor of, 324–325, 372 multiplication of, 288 polynomials divided by, 305–306, 314 polynomials multiplied by, 289 Multiplication associative property of, 59–60, 78, A:16 of binomials, 294–298, A:38–A:39 in building up fractions, 14 commutative property of, 58, 77, A:16 in factoring out greatest common factor, 326 using, 66–67 of complex numbers, 608–609 of conjugates, 582, 610 distributive property of, A:16 over addition See Distributive property of multiplication over addition of exponential expressions, 256 of fractions, 15–16, 76, A:13 vs addition, 410 function for, 752, 773 identity property of, 62, 78, A:16 inverse property of, 62–63, 78, A:16 of monomials, 288 with polynomials, 289 in order of operations, 43, 44 of polynomials, 289–290, 313, A:38 applications of, 290–291 by monomials, 288 of radicals with different indices, 582–583 with same index, 580–582 of rational expressions, 393–394, 447, A:52–A:53 of rational numbers, 392–393 of real numbers, 34–35, 77, A:14–A:15 solving linear equalities in one variable by, 88–90, 94–95 symbols of, 34, 51 verbal expressions for, 49, 50, 121 Multiplication properties of equality, 88–90, 94–95, 160, A:20–A:21 applications of, 91 simplifying before using, 97–98, 103 of inequality, 152, 153–156, 161 of zero, 63, 78, A:16 Multiplicative identity, 62, 78 Multiplicative inverse, 62–63, 78 N Natural base, 789 Natural logarithm, 802, 829 Natural numbers, 2, 76, A:2, A:11 Negative exponents, 264–272, 312, A:42 definition of, 268 integral, 264–267, 312 rational, 570–572 rules for, 265, 267–269, 312, A:42 for exponential expressions, 265 Negative numbers absolute value equal to, 519–520 addition of, 26–27 even roots of, 574 in exponential expressions, 42–43 in long division, 36 in quadratic formula, 641 in set of integers, square roots of, 611–612 Negative numbers, in set of integers, Negative rational exponents, 570–572 Negative sign (Ϫ) for opposites, in trial and error method, 350 Negative slope, 188–189, 242 Net worth, 27, 28, 30, 31, 145 Nonlinear equations, 840 Nonlinear systems of equations, 840–849, 887 applications of, 843–845 with conical sections, 864 graphing, 840 with logarithms, 842–843 solving by addition method, 841–842 by elimination, 840–843 by substitution, 841 Notation exponential, 41 function, 177, 696–697, 771, 774 for inequalities, 145 interval, 6–8, 76, A:11–A:12 radical, 558–559, 616 scientific See Scientific notation set-builder, 3, A:2–A:3 nth roots, 558, 616 Number(s) complex See Complex numbers composite, 322 expressed as numbers and words, 275 imaginary See Imaginary numbers integers See Integers irrational See Irrational numbers mixed, 18 natural See Natural numbers I-7 negative See Negative numbers pairs of, algebraic expressions for, 121–123, 124 prime, 19, 322, 372 rational See Rational numbers real See Real numbers sets of, whole, 2, 76 Number line, 3–4 comparing numbers on, coordinates on, graphing on, inequalities on, 144, 145 rational numbers on, real numbers on, zero (origin) on, Number problems, 130 Numerator, 13 O Oblique asymptote, of rational function, 740 Odd root, 558 Odd-root property, 596 One-to-one function, 761, 773 exponential functions as, 792–793 logarithmic functions as, 804–805 One-to-one property of exponential functions, 793 of logarithms, 804–805, 820–821 Open circle (symbol), Operations with functions, 751–753 order of See Order of operations Opposites See also Additive inverse of common factor, factoring out, 387–388 of greatest common factor, factoring out, 326–327 of an opposite, 10 of real numbers, 9–10, 28, 62, 77 removing parentheses with, 71 “Or,” in compound inequalities in one variable, 508, 547 in two variables, 528–529, 530–531, 533–534, 547 Ordered pairs See also Point(s) compound inequalities in two variables satisfied by, 528–529 definition of, 170 finding, 658–659 functions expressed by, 693–694 graphing, 170–171 linear equations in two variables satisfied by, 171–172, 231–232 relations as, 693 Ordered triples, 487 Order of operations, 43–45, 77, A:15 applications of, 45–46 grouping symbols and, 40–41, 43, 44–45 memory device for, 43 in solving linear equations in one variable, 95 dug84356_index.qxd 10/1/10 2:49 PM Page I-8 I-8 Index Origin in coordinate plane, 170 ellipse centered at, 869 on number line, symmetry about, 728, 772 as test point, 235 P Pairs of numbers See also Ordered pairs algebraic expressions for, 121–123, 124 Parabolas, 849–861, 887–888 axis of symmetry of, 853–854 definition of, 851 directrix of, 851 finding, 853 equations of, 851–852, 887 changing form of, 854–855 in form x ϭ a(y Ϫ k)2 ϩ h, 855–856, 887 in form x ϭ ay2 ϩ by ϩ c, 888 in form y ϭ a(x Ϫ h)2 ϩ k, 852–855, 887 in form y ϭ ax2 ϩ bx ϩ c, 854–855, 888 focus of, 851 finding, 853 as graph of function, 712 of quadratic function, 659–660, 679, 704 of second-degree inequality, 881 graphing, 856 intercepts of, 661–662 maximum value of, 660 minimum value of, 660 in nonlinear system of equations, 840 opening downward, 852 opening left, 855–856 opening right, 855–856 opening upward, 852 properties of, 679 vertex of, 660–661, 851 finding, 853 Parallel lines, 190–191 definition of, 190 graphing, 190–191 point-slope form with, 213–214 slope of, 190–191, 192, 242 Parallelogram, area of, 125 Parentheses with algebraic expressions, 51 in combining sets, A:5–A:6 on graphing calculator, with fractional divisor, 18 in interval notation, 6, in linear equations in one variable, 97–98 and order of operations, 40, 43, 44, 45 removing, 70–72, 98 PEMDAS, 43 Percentage models, 124 Percents converting decimals and fractions to, 21 converting to decimals and fractions, 21 fractions as, 21 Percent symbol (%), 21 Perfect cubes, 559 Perfect fourth powers, 559 Perfect square(s), 333, 559 Perfect square trinomials in completing the square, 629–630 factoring, 333–335, 372, 629–631, A:45–A:46 identifying, 334 last term of, 629–630 in quadratic equations, 363–364 Perimeter applications of, 72, 131 formula for, 114–115 geometric models for, 125 of rectangle, 72, 114–115, 125 of square, 125 Perpendicular lines, 191–192 definition of, 191 graphing, 191–192 slope of, 191–192, 242 with point-slope form, 214–215 Pi (␲), as irrational number, Piecewise functions, 706 Place value system, 21 “Please Excuse My Dear Aunt Sally” (PEMDAS), 43 Plotting See Graphing Plotting points, in rectangular coordinate system, 170–171 Point(s) finding slope of line through two, 213 plotting in rectangular coordinate system, 170–171 Point-slope form, 211–223, A:32 applications of, 215–216 definition of, 212, 242 with parallel lines, 213–214 with perpendicular lines, 214–215 vs slope-intercept form, 212 writing equations in, 212–213 Poiseuille’s law, 255 Polynomial(s), 279–281, 313, A:37–A:38 See also Binomial(s); Monomials; Trinomials addition of, 282, 283, 313 applications of, 284 additive inverse of, 290 applications of, 284, 290–291 definition of, 279–280, 313, A:37 degree of, 280, 313, A:37 division of, 305–311, 314, A:40 by binomials, 306–309, 314 by monomials, 305–306, 314 evaluating, 281–282, 313 results of, 282 factoring See also Factoring completely, 335–336, A:49 four-term, factoring by grouping, 330–332 irreducible, 335 least common denominator for, strategy for finding, 403 multiplication of, 289–290, 313, A:38 applications of, 290–291 naming, 313 prime, A:49 definition of, 335, 372 factoring, 342–343 sum of two squares, 343 ratio of two See Rational expressions subtraction of, 283, 313 applications of, 284 types of, 280–281 Polynomial functions definition of, 281–282, 772 graphing, 725–738 behavior at x-intercepts, 728–729, 772 cubic functions, 725–726 quartic functions, 726–727 symmetry in, 727–728, 772 transformations of, 730, 772 Polynomial inequalities, 772 solving, 730–732, 772 Positive numbers, absolute value equal to, 519–520 Positive slope, 188–189, 242 Power(s) See also Exponent(s) of 2, 266 of binomials, 301 eliminating from equation, 601 of imaginary numbers, 609 of radical expressions, 592–593 raising each side of equation to, 598–600 in rational exponents, 571 of rational expressions, 592–593 in scientific notation, 273 Power key, 42 Power of a power rules for exponents with negative exponents, 269 with positive exponents, 258–259, 312, A:41 for rational exponents, 572–573, 617 Power of a product rules for exponents with negative exponents, 269 with positive exponents, 259–260, 312, A:41 for rational exponents, 572–573, 617 Power of a quotient rules for exponents with negative exponents, 269 with positive exponents, 260, 312, A:41 for rational exponents, 572–573, 617 Power of ten See Scientific notation Power rules for logarithms, 813–814, 829 Predictor variable See Independent variable Present value formula, 269–270 Prime factorization, 322–323 Prime numbers, 19, 322, 372 Prime polynomials, A:49 definition of, 335, 372 factoring, 342–343 sum of two squares, 343 Prime quadratic polynomials, 650 Principal, 261 Principal roots, 558, 616 Problem solving, A:24–A:25 Product See also Multiplication as algebraic expression, 49, 50 definition of, 34, A:14 of a difference, 314 finding, 69 of radicals, 587 special, 299–305, A:39–A:40 and conjugates, 582 identifying, 334 of a sum, 314 of a sum and a difference, 300, A:39–A:40 Product function, 752, 773 Product rules for exponents, A:38, A:41 with negative exponents, 268–269 with polynomials, 313 with positive exponents, 256, 312 for logarithms, 812, 819–820, 829 for radicals, 560–562, 563, 587, 616 in simplifying radicals, 587 for rational exponents, 572–573, 617 Proportion(s), 431–434, 448, A:33 applications of, 433–434, A:55–A:56 definition of, 431, A:55 solving with extremes-means property, 432–433, 448 Proportionality constant See Variation constant Pythagorean theorem, 366–367 Q Quadrants, 170 Quadratic equations See also Parabolas applications of, 653–654 definition of, 361, 678, A:49 discriminant of, 644, 650 from equations with radicals, 633–634 factoring, 361–371, 628, 678 applications of, 366–367 factors of, correspondence with solutions, 649–650 form of, 628 formula for See Quadratic formula fractions in, converting to integers, 365–366 perfect square trinomials in, 363–364 solutions to compound equations, 362 correspondence with factors, 649–650 given, 649–650 imaginary, 634–635 irrational, 632–633 number of, 364, 678 rational, 631–632 solving, 628–638, 678 by completing the square, 629–633, 643, 678 dug84356_index.qxd 10/1/10 2:49 PM Page I-9 Index by even-root property, 629, 643, 678 by factoring, 361–371, 373, 628, 643, 678, A:49–A:50 methods for, 643 by quadratic formula See Quadratic formula by zero factor property, 361 writing, 678 with given solutions, 649–650 zero factor property of, 361–364, 373, A:49 Quadratic family of functions, 712 Quadratic form, 650–652, 678 Quadratic formula, 639–648, 678 applications of, 644–645 definition of, 640 developing, 639–640 discriminant of, 644 solutions to imaginary, 643 irrational, 642 number of, 643–645 rational, 641–642 using, 641–643 Quadratic functions definition of, 658, 704, 771 graphing, 658–668, 679, 704 ordered pairs for, 658–659 Quadratic inequalities, 668–677, 679 applications of, 673 definition of, 668–669 solving with graphical method, 668–671, 679 with test-point method, 671–673, 679 Quadratic polynomials factoring, 678 prime, 650 Quartic functions, 726–727 Quotient, 18, 49, 50 See also Division definition of, A:14 of division of polynomials, 307, A:40 simplifying, 69–70 undefined, 37 Quotient function, 752, 773 Quotient rule for exponents, 312, A:40, A:41 with negative exponents, 268–269 with positive exponents, 257–258 reducing rational expressions with, 386–387 Quotient rule for logarithms, 813, 829 Quotient rule for radicals, 562–563, 617 in simplifying radicals, 588 Quotient rule for rational exponents, 572–573, 617 R Radical(s), 558–567 addition of, 579–580 division of, with same index, 590–592 equations with, 596–606, 617 applications of, 602–603 quadratic equations from, 633–634 evaluating, 558–559 index of, 558 like, 579 multiplication of with different indices, 582–583 with same index, 580–582 parts of, 558 product rule for, 560–562, 563, 587, 616 quotient rule for, 562–563, 617 rationalizing denominator of, 592 simplified form for, 588–589, 617 simplifying, 561–562, 587–589 before combining, 579–580 before division, 591 with product rule, 587 with quotient rule, 588 subtraction of, 579–580 Radical expressions domain of, 563–564, 616 powers of, 592–593 simplifying, 591–592 Radical functions, 564, 616 Radical notation, 558–559, 616 Radical symbol (͙), 558 Radicand, 558 Radius of circle, 861 Range in composition of functions, 754 of function, 695–696, 701, 771 of inverse function, 760 of logarithmic functions, 801, 802–803 of relation, 695–696 Rate, distance, and time See Distance, rate, and time Rate of discount, 115, 137 Rate problems, 442–443 See also Work problems rational expressions in, 388 Ratio(s), 429–430 See also Proportion(s) applications of, 430–431, A:55–A:56 equivalent, 430 as fractions, 430 rational numbers as, 2–3 rational numbers used for, undefined, units in, 431 Rational exponents, 568–578, 616 definitions of, 568 denominator of, 568 in equations quadratic in form, 652 equations with, 601–602 evaluating, 570–571 in expressions involving variables, simplifying, 573–575 negative, 571–572 numerator of, 568 rules of exponents for, 572–573, 617 Rational expressions, 381–455, A:52–A:58 addition of, 409–412, 447, A:53–A:54 applications of, 438–446, A:56 addition, 412 rates, 388, 396–397 building up denominators in, 401–402 in complex fractions See Complex fractions converting to least common denominator, 403–404 definition of, 382, 447, A:52 division of, 394–396, 447, A:52–A:53 domain of, 383–384 equations with extraneous solutions to, 426–427 quadratic equations from, 634 solving, 424–429, 448, A:55 with two solutions, 425–426 variables in denominators of, 425 evaluating, 382 least common denominator of, 400–407 converting to, 403–404 finding, 402–403, 447 in lowest terms, 384–386, A:52 multiplication of, 393–394, 447, A:52–A:53 powers of, 592–593 reducing, 384–386, 447, A:52 by dividing a Ϫ b by b Ϫ a, 387 with quotient rule for exponents, 386–387 strategy for, 388 ruling out values for, 383 subtraction of, 409–412, 447, A:53–A:54 undefined, 383 writing, 388 Rational functions asymptotes of, 738–741, 773 definition of, 382, 447, 738, 772 domain of, 384, 738 graphing, 738–751 sketching, 741–742 Rational inequalities, 743–745, 773 solving by graphing method, 743, 773 by test-point method, 743–744, 773 Rationalizing denominators, 586–587, 592 Rational numbers, 2–3 See also Fraction(s) addition of, 407–409 converting to decimals, definition of, A:11 division of, 394 multiplication of, 392–393 on number line, as quadratic equation solutions, 631–632 set of, 2–3, 76 subtraction of, 407–409 vinculum in, 393 Real estate commission model, 125 I-9 Real numbers, 1–84, A:11–A:19 absolute value of, 9–10, 27, 28, 77 absolute value inequalities solved by, 523 addition of, 26–29, A:13–A:14 applications of, 31 with like sign, 26–27, 77 with unlike signs, 27–29, 77 associative properties of, 59–60, 78 commutative properties of, 58, 77 as complex numbers, 608 definition of, 5, A:11–A:12 distributive property of multiplication over addition for, 60–62, 78 division of, 35–37, 77, A:14–A:15 with like sign, 35, 36, 37 with unlike signs, 35, 36, 37 identity properties of, 62, 78 intervals of, 6–8, 76, A:11 inverse properties of, 28, 62–63, 78 multiplication of, 34–35, 77, A:14–A:15 opposites of, 9–10, 28, 62, 77 properties of, A:16–A:18 reciprocal of, 62 set of, 5, 76 subtraction of, 29–31, 77, A:13–A:14 Real part of complex numbers, 607 Reciprocals in rational exponents, 571 of real numbers, 62 Rectangle area of, 125 perimeter of, 72, 114–115, 125 Rectangular coordinate system, 170 See also Graphing changing scale on, 175–176 plotting points in, 170–171 slope in, 185 Reducing vs building up, 402 fractions, 76, A:12–A:13 rational expressions, 384–386, 447, A:52 by dividing a Ϫ b by b Ϫ a, 387 in multiplication of, 393 with quotient rule for exponents, 386–387 strategy for, 388 Reducing fractions, 15 Reflection of exponential functions, 792 of functions, 714–715 Regression line, 184 Relations definition of, 693, 771 domain of, 695–696 as equations, 694 graphing, 706–707 range of, 695–696 Repeating decimal, Resistance, 439 Response variable See Dependent variable Right, translation to, 712 Right-opening parabola, 855–856 Right triangle See Pythagorean theorem dug84356_index.qxd 10/1/10 2:49 PM Page I-10 I-10 Index Rise, 185, A:29 Roots, 558–559 absolute value with, 559–560, 573–574 eliminating from equation, 601 to equation See Solutions, to equations even, 558 of exponential expressions, 560 with exponents, 559–560 extraneous, 426–427 nth, 558, 616 odd, 558 principal, 558, 616 in rational exponents, 571 square See Square root Rules for rational exponents, 572–573 Run, 185, A:29 S Satisfying equations See Solutions, to equations Scientific notation, 273–278, 313, A:43 applications of, 276 computations with, 275–276, A:43 converting standard notation to, 274, 313, A:43 converting to standard notation, 273–274, 313, A:43 definition of, 273 rules of exponents with, 275–276 words converted to, 275 Second-degree inequalities, 881–886, 889–890 definition of, 881 graphing, 881–882 solution sets to, 882, 889 systems of, 882, 890 Seesaw, 152 Selling price and discount model, 124 Set(s) combining, A:5–A:6 definition of, A:2 empty, 106, A:4 equal, A:3 of integers, 2, 76 intersection of for compound inequalities in one variable, 509, 512, 547 for compound inequalities in two variables, 529–530, 547 definition of, A:4 of natural numbers, 2, 76 of numbers, of rational numbers, 2–3, 76 of real numbers, 5, 76 solution See Solution set(s) subsets, A:5 union of for compound inequalities in one variable, 510, 512, 547 for compound inequalities in two variables, 530–531, 533–534, 547 definition of, A:3 of whole numbers, 2, 76 Set-builder notation, 3, A:2–A:3 Shrinking, of functions, 713–714, 772 Similar triangles, 186 Simple inequalities, 508 Simple interest, 114 Simplification of algebraic expressions, 71–72, A:17 of complex fractions, 418, 448 strategy for, 418 using least common denominator, 418–420 definition of, 68 of expressions, 66–75 of linear equations in one variable, 97–98, 103 of quotients, 69–70 of radical expressions, 591–592 of radicals, 561–562, 587–589 before combining, 579–580 before division, 591 with product rule, 587 with quotient rule, 588 of rational exponent expressions with variables, 573–575 Simplified radical form, 588, 617 Slant asymptote, of rational function, 740 Slash (͞), 51 Slope-intercept form, 199–210, A:30–A:31 applications of, 204–205 in classifying systems of linear equations, 461 definition of, 199–200, 242 vs point-slope form, 212 writing equations in, 203–204 Slope of line, 185–198, 242, A:29–A:30 applications of, 192–193 coordinate formula for, 187–189, 204 definition of, 185 finding, 185, 204, A:30 through two points, 213 using coordinates for, 187–189, 204 using slope-intercept form, 200 graphing line from, 189–190 of horizontal lines, 188, 242 negative, 188–189, 242 of parallel lines, 190–191, 192, 242 of perpendicular lines, 191–192, 214–215, 242 positive, 188–189, 242 undefined, 188, 242 of vertical lines, 188, 190, 191, 242 and y-intercept See Slope-intercept form zero, 188, 242 Solutions to absolute value inequalities all real numbers, 523 no real numbers, 524 to equations, 52, 86, 87, A:20 extraneous, 426–427, 599 imaginary, 612–613 to linear equations in two variables, ordered pairs as, 171–172 to quadratic equations See Quadratic equations, solutions to to quadratic formula imaginary, 643 irrational, 642 number of, 643–645 rational, 641–642 to systems of linear equations See Systems of linear equations Solution set(s) to absolute value inequalities, 521 to compound inequalities in one variable, 508–509 graphing, 509–513 intersection of, 509, 512 overlapping intervals, 510–511 union of, 510, 512 to compound inequalities in two variables, graphing, 529–530 definition of, 86 to equations, A:20 to identities, 105, 107 to inconsistent equations, 106, 107 to inequalities, 145 to linear inequalities in two variables, 232 to second-degree inequalities, 881, 889 to systems of second-degree inequalities, 882, 890 Special products, 299–305, A:39–A:40 and conjugates, 582 identifying, 334 Square (geometric figure) area of, 125 perimeter of, 125 Square(s) of binomials, 630–631 completing See Completing the square of a difference, 300, 314, A:39–A:40 difference of factoring, 332–333, 372, A:45–A:46 as product of a sum and a difference, 300 perfect, 333, 559 in second-degree inequalities, 881 of a sum, 299, 314, A:39–A:40 visualizing, 299 sum of, prime, 343 Square family of functions, 712 Square root See also Irrational numbers definition of, 558 as irrational number, of negative numbers, 611–612 of x2, 573 Square-root family of functions, 712 Square-root functions, 705, 771 Standard form equation of a circle in, 862–863 of linear equations in two variables, 200–201, 203, 242, A:30–A:31 applications of, 204–205 used in accounting, 482 Standard notation converting scientific notation to, 273–274, 313, A:43 converting to scientific notation, 274, 313, A:43 words converted to, 275 Standard viewing window, 189 Star (*), 51 Stretching, of functions, 713–714, 772 Subscripts, 187 Subsets, A:5 Substitution nonlinear systems of equations solved by, 841 systems of linear equations solved by in three variables, 489, 497 in two variables, 467–468, 482, 497 Subtraction in complex fractions, 419 of complex numbers, 608 of fractions, 18–21, 76, 407–409, A:13 with same denominator, 408 function for, 752, 773 in long division, 308 in order of operations, 43, 44 of polynomials, 283, 313 applications of, 284 of radicals, 579–580 of rational expressions, 409–412, 447, A:53–A:54 of rational numbers, 407–409 of real numbers, 29–31, 77, A:13–A:14 removing parentheses with, 71 solving linear equations in one variable by, 87 verbal expressions for, 49, 50, 120 Sum, 49, 50 See also Addition product of, 314 product of difference and, 300, A:39–A:40 square of, 299, 314, A:39–A:40 of two cubes, factoring, 355–356, 372, A:48–A:49 of two squares, prime, 343 Sum function, 752–753, 773 Supplementary angles, 122–123 Supply, 462 Switch-and-solve strategy, 763–765, 774 Symbols absolute value bars, 9, 40, 44, 45 approximately equal to (Ϸ), braces, 2, 87 brackets, 6, 44, 45 caret (^), 51 closed circle, equality (ϭ), 52 fraction bars, 37, 40, 44 greater than (Ͼ), 144 greater than or equal to (у), 144 grouping, 40–41, 43, 44–45 inequality, 144, A:26 infinity (ϱ), 7, 145 for irrational numbers, less than (Ͻ), 144 less than or equal to (р), 144 multiplication, 34, 51 negative sign (Ϫ), open circle, dug84356_index.qxd 10/1/10 2:49 PM Page I-11 Index opposite (Ϫ), percent (%), 21 radical (͙), 558 slash (͞), 51 star (*), 51 Symmetry, of graphs of functions, 727–728, 772 Systems of linear equations, 457–506 in three variables, 487–496, 497 applications of, 491–492 consistent, 490 definition of, 487 dependent, 490–491 inconsistent, 490–491 independent, 490 with infinitely many solutions, 491 with no solutions, 491 with single solution, 488 solving by addition, 497 solving by addition and substitution, 489 solving by elimination, 487–490, 497 solving by substitution, 497 strategy for solving, 489 in two variables, 497 applications of, 462, 470, 482–483 consistent See Consistent systems of linear equations, in two variables with decimals, 481–482 dependent See Dependent systems of linear equations, in two variables with fractions, 480–481 inconsistent See Inconsistent systems of linear equations, in two variables independent See Independent systems of linear equations, in two variables with infinitely many solutions, 460 with no solution, 460–461 with one solution, 459–460 solution to, 458 solving by addition method, 477–487, 482, 497 solving by graphing, 458–461, 482, 497 solving by substitution, 467–468, 482, 497 strategy for solving, 468, 479 types of, 461–462, 497 Systems of nonlinear equations See Nonlinear systems of equations Systems of second-degree inequalities, 882, 890 T Table(s), functions expressed by, 692–693 TABLE feature on graphing calculator, 153 Term(s) constant, 280 definition of, 67, 279, 313, A:17 degree of, 280 last, finding, 629–630 like See Like terms TEST menu on graphing calculator, 144 Test-point method for compound inequalities in two variables, 530, 531 for linear inequalities in one variable, 236 for linear inequalities in two variables, 235, 243 for polynomial inequalities, 731–732, 772 for quadratic inequalities, 671–673, 679 for rational inequalities, 743–745, 773 Thermometer, Time, distance, rate and See Distance, rate, and time Transformation of exponential functions, 791–792 of functions, 712–723, 772 horizontal translation, 712–713, 772 multiple, 716–718 reflection, 714–715 stretching and shrinking, 713–714, 772 vertical translation, 715–716, 772 of polynomial functions, 730 Translation (transformation) of exponential functions, 792 of functions horizontal, 712–713, 772 vertical, 715–716, 772 Translations of verbal expressions to algebraic form, 49–51 involving addition, 49, 50, 120 involving division, 49, 50, 121 involving linear equations, 52–53, 120–129 involving multiplication, 49, 50, 121 involving subtraction, 49, 50, 120 Trial and error method, for factoring trinomials, 349–351, 373, A:48 Triangles See also Pythagorean theorem area of, 125 degree measures of angles, 122–123 hypotenuse of, 367 similar, 186 Trinomials See also Polynomial(s) ax2 ϩ bx ϩ c with a ϭ 1, factoring, 339–346 definition of, 280, A:37 factoring ac method for, 347–349, 350, 372 ax2 ϩ bx ϩ c with a ϭ 1, 339–346 ax2 ϩ bx ϩ c with a 1, 347–358 completely, 343, 351–352, 373 by grouping, 339 perfect square trinomials, 333–335, 372 trial and error method for, 349–351, 373 with two variables, 343 perfect square factoring, 333–335, 372 identifying, 334 in quadratic equations, 363–364 U Unbounded (infinite) interval, 7–8 Unbounded intervals, 7–8 Undefined quotient, 37 Undefined ratio, Undefined rational expressions, 383 Undefined slope, 188, 242 Uniform motion applications, 132–133, A:24–A:25 formulas for, 438, 439–440 rational expressions in, 388, 396–397 Uniform motion model, 124 See also Distance, rate, and time Union method, 530–531 Union of sets (ʜ) for compound inequalities in one variable, 510, 512, 547 in two variables, 530–531, 533–534, 547 definition of, A:3 Unit(s) canceling, 17 conversion of, 17 on number line, in ratios, 431 Universal product codes (UPC), 760 Upward-opening parabola, 659–660, 852 Upward translation, 716 V Variables on both sides of equation, 90–91, 112–113 coefficient of See Coefficient definition of, dependent, in functions, 171–172, 691, 693 finding value of, in formula, 113, 439 independent, in functions, 171–172, 691, 693 isolating, on one side with addition property of equality, 86–88, 90–91, 95–97 with addition property of inequality, 153–156 for equations of form ax ϩ b ϭ cx ϩ d, 95–97 with multiplication property of equality, 88–90 with multiplication property of inequality, 153–156 number preceding See Coefficient I-11 rational exponent expressions with, simplifying, 573–575 rewriting formulas for, 110–113, 160, A:23 in sets, A:2 solving for certain, 110–111, 160, A:23 uses of, 49 Variation, 223–230, 243, A:33 applications of, 226–227 determining form of, 226, 227 direct, 223–224, 243, A:33 inverse, 224–225, 243, A:33 joint, 225, 243, A:33 Variation constant, 223, 224, A:33 finding, 225–226 Venn diagrams, A:3 Verbal expressions, translating to algebraic form, 49–51, 120–129, A:23 involving addition, 49, 50, 120 involving division, 49, 50, 121 involving inequalities, 147 involving linear equations, 52–53, 120–129 involving multiplication, 49, 50, 121 involving subtraction, 49, 50, 120 words used for, 120–121 Verbal problems See also Applications strategy for solving, 130–131 Vertex of parabola, 660–661, 851 finding, 853 Vertical asymptote, of rational function, 738–741 Vertical boundary lines, 234 Vertical lines graphing, 175 slope of, 188, 190, 191, 242 Vertical-line test, 694–695, 771 accuracy of, 762 Vertical translation, 715–716, 772 Viewing window, of graphing calculator, 189, 204 Vinculum, 393 W Whole numbers, 2, 76, A:11 Words for algebraic expressions, 120–121 numbers expressed with, 275 Work problems rational expressions in, 388, 397, 440–441 strategy for solving, 441 X x-axis as asymptote of exponential function, 790 changing scale on, 175–176 definition of, 170 dug84356_index.qxd 10/1/10 2:49 PM Page I-12 I-12 Index x-coordinate definition of, 170 of exponential function, 794 in slope, 185 x-intercept behavior of polynomial functions at, 728–729, 772 cover-up method for finding, 176 of cubic function, 725–726 definition of, 176, A:29 graphing line using, 176–177, A:29 of parabola, 661–662 of quartic function, 726–727 xy-plane See Coordinate plane Y y-axis changing scale on, 175–176 definition of, 170 parabola symmetric about, 853–854 symmetry about, 727, 772 y-coordinate definition of, 170 in slope, 185 y-intercept cover-up method for finding, 176 definition of, 176, A:29 of exponential functions, 790 finding, 200 graphing line using, 176–177, A:29 of parabola, 661–662 in slope-intercept form, 199 and slope of line See Slope-intercept form Z Zero absolute value equal to, 519–520 division by, 37 division of, 36, 37 as exponent, 257, 312, A:41 multiplication property of, 63, 78 on number line, opposite of, Zero factor property, 361–364, 373, 628, A:49 Zero slope, 188, 242 ... Vancomycin dosage, 119 Waist-to-hip ratio, 539 Weight of dogs, 502 dug8 4356_fm.indd xxvi Weight of twins, 526–527 Weights of three people, 495 Weights of two people, 473 Winter wheat, 371 Business... Elementary and Intermediate Algebra F o u r t h e d i t i o n Mark Dugopolski Southeastern Louisiana University TM dug8 4356_fm.indd i 10/28/10 7:01 PM TM ELEMENTARY AND INTERMEDIATE ALGEBRA, FOURTH EDITION... edition) (hard copy: alk paper) Algebra Textbooks I Title QA152.3.D84 2012 512.9—dc22 2010024307 www.mhhe.com dug8 4356_fm.indd ii 10/28/10 7:01 PM About the Author M ark Dugopolski was born and raised