Introductory & Intermediate Algebra for College Students
This page intentionally left blank
Introductory & Intermediate Algebra for College Students
Annotated Instructor’s Edition
Robert Blitzer
Miami Dade College
Editorial Director: Christine Hoag
Senior Acquisitions Editor: Dawn Giovanniello
Executive Content Editor: Kari Heen
Associate Content Editor: Katherine Minton
Editorial Assistant: Chelsea Pingree
Senior Managing Editor: Karen Wernholm
Senior Production Project Manager: Kathleen A. Manley
Digital Assets Manager: Marianne Groth
Media Producer: Shana Siegmund
Software Development: Eric Gregg, MathXL; and Mary Durnwald, TestGen
Executive Marketing Manager: Michelle Renda
Marketing Assistant: Susan Mai
Senior Author Support/Technology Specialist: Joe Vetere
Rights and Permissions Advisor: Michael Joyce
Image Manager: Rachel Youdelman
Procurement Manager: Evelyn Beaton
Procurement Specialist: Debbie Rossi
Senior Media Buyer: Ginny Michaud
Associate Director of Design: Andrea Nix
Senior Designer: Beth Paquin
Text Design: Ellen Pettengell Design
Production Coordination: Rebecca Dunn/Preparé, Inc.
Composition: Preparé, Inc.
Illustrations: Scientific Illustrators/Laserwords
Cover Design and Illustration: Nancy Goulet, studio;wink
For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C1, which is hereby made part of this copyright page.
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Blitzer, Robert.
Introductory & Intermediate algebra for college students / Robert Blitzer. —4th ed. p.cm.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-6713447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—QGD—15 14 13 12
ISBN 10: 0-321-75894-3
ISBN 13: 978-0-321-75894-1 www.pearsonhighered.com
Preface ix
To the Student xvii
About the Author xix
Table of Contents
1.2 Fractions in Algebra 14
1.3 The Real Numbers 32
1.4 Basic Rules of Algebra 44
1.5 Addition of Real Numbers 56
1 Variables, Real Numbers, and Mathematical Models1
1.1 Introduction to Algebra: Variables and Mathematical Models 2
IntroductoryandIntermediateAlgebra for College Students, Fourth Edition, provides comprehensive, in-depth coverage of the topics required in a course combining the study of introductory and intermediate algebra. The book is written for college students who have no previous experience in algebra and for those who need a review of basic algebraic concepts before moving on to intermediate algebra. I wrote the book to help diverse students, with different backgrounds and career plans, to succeed in a combined introductory and intermediate algebra course. Introductory andIntermediateAlgebra for College Students, Fourth Edition, has two primary goals:
1.To help students acquire a solid foundation in the skills and concepts of introductory and intermediate algebra, without the repetition of topics in two separate texts.
2.To show students how algebra can model and solve authentic real-world problems.
One major obstacle in the way of achieving these goals is the fact that very few students actually read their textbook. This has been a regular source of frustration for me and for my colleagues in the classroom. Anecdotal evidence gathered over years highlights two basic reasons why students do not take advantage of their textbook: “I’ll never use this information.”
“I can’t follow the explanations.”
I’ve written every page of the Fourth Edition with the intent of eliminating these two objections. The ideas and tools I’ve used to do so are described in the features that follow. These features and their benefits are highlighted for the student in “A Brief Guide to Getting the Most from This Book,” which appears inside the front cover.
What’s New in the Fourth Edition?
New Applications and Real-World Data. I’m on a constant search for data that can be used to illustrate unique algebraic applications. I researched hundreds of books, magazines, newspapers, almanacs, and online sites to prepare the Fourth Edition. Among the worked-out examples and exercises based on new data sets, you’ll find applications involving stretching one’s life span, the emotional health of college freshmen, changing attitudes toward marriage, American Idol’s viewership, and the year humans become immortal.
Concept and Vocabulary Checks. The Fourth Edition contains more than 1500 new short-answer exercises, mainly fill-in-the-blank and true/false items, that assess students’ understanding of the definitions and concepts presented in each section. The Concept and Vocabulary Checks appear as separate features preceding the Exercise Sets.
Great Question! This feature takes the content of each Study Tip in the Third Edition and presents it in the context of a student question. Answers to questions offersuggestions for problem solving, point out common errors to avoid, and provideinformal hints and suggestions. “Great Question!” should draw students’ attention and curiosity more than the “Study Tips.” As a secondary benefit, this new feature should help students not to feel anxious or threatened when asking questions in class.
Achieving Success. The book’s Achieving Success boxes offer strategies for success in math courses, as well as suggestions for developmental students on achieving one’s full academic potential.
New Chapter-Opening and Section-Opening Scenarios. Every chapter and every section open with a scenario based on an application, many of which are unique to the Fourth Edition. These scenarios are revisited in the course of the chapter or section in one of the book’s new examples, exercises, or discussions. The often-humorous tone of these openers is intended to help fearful and reluctant students overcome their negative perceptions about math.
New The Lecture Series on DVD has been completely revised to provide students with extra help for each section of the textbook. The Lecture Series DVD includes: Interactive Lectures that highlight key examples and exercises for every section of the textbook. A new interface allows easy navigation to sections, objectives, and examples.
Chapter Test Prep Videos provide step-by-step solutions to every problem in each Chapter Test in the textbook. The Chapter Test Prep Videos are now also available on You-Tube™.
What Content Changes Have Been Made to the Fourth Edition?
Section 2.3 (Solving Linear Equations) contains a new discussion on solving linear equations with decimals.
Section 2.4 (Formulas and Percents) speeds up the discussion on basics of percent, a prealgebra topic, with the focus on algebraic applications.
Section 2.7 (Solving Linear Inequalities) includes an additional application involving an example about staying within a budget.
Section 6.1 (The Greatest Common Factor and Factoring by Grouping) includes a new objective on factoring out the negative of a polynomial’s greatest common factor.
Section 6.5 (A General Factoring Strategy) slows down the pace of the Practice Exercises in the Exercise Set. The Exercise Set now opens with problems that require students to use only one factoring technique before moving on to multiple-stepfactorizations.
Section 7.6 (Solving Rational Equations) expands the coverage of solving a formula with a rational expression for a variable with a new worked example and Check Point.
Section 9.3 (Equations and Inequalities Involving Absolute Value) solves absolute value inequalities using equivalent compound inequalities, rather than boundary points. With the section’s focus on this one method, the pace has been both slowed down and expanded.
Section 11.1 (The Square Root Property and Completing the Square) contains a new example on solving a quadratic equation with imaginary solutions by completing the square.
Section 11.2 (The Quadratic Formula) has a new example on writing a quadratic equation whose solution set contains irrational numbers.
Section 11.5 (Polynomial and Rational Inequalities) includes a new example on solving a polynomial inequality with irrational boundary points.
What Familiar Features Have Been Retained in the Fourth Edition?
Detailed Worked-Out Examples. Each worked example is titled, making clear the purpose of the example. Examples are clearly written and provide students with detailed step-by-step solutions. No steps are omitted and key steps are thoroughly explained to the right of the mathematics.
Explanatory Voice Balloons. Voice balloons are used in a variety of ways to demystify mathematics. They translate algebraic ideas into everyday English, help clarify problem-solving procedures, present alternative ways of understanding concepts, and connect problem solving to concepts students have already learned.
Check Point Examples. Each example is followed by a similar matched problem, called a Check Point, offering students the opportunity to test their understanding of the example by working a similar exercise. The answers to the Check Points are provided in the answer section.
Extensive and Varied Exercise Sets. An abundant collection of exercises is included in an Exercise Set at the end of each section. Exercises are organized within eight category types: Practice Exercises, Practice Plus Exercises, Application Exercises, Writing in Mathematics, Critical Thinking Exercises, Technology Exercises, Review Exercises, and Preview Exercises. This format makes it easy to create well-rounded homework assignments. The order of the Practice Exercises is exactly the same as the order of the section’s worked examples. This parallel order enables students to refer to the titled examples and their detailed explanations to achieve success working the Practice Exercises.
Practice Plus Problems. This category of exercises contains more challenging practice problems that often require students to combine several skills or concepts. With an average of ten Practice Plus problems per Exercise Set, instructors are provided with the option of creating assignments that take Practice Exercises to a more challenging level.
Mid-Chapter Check Points. At approximately the midway point in each chapter, an integrated set of Review Exercises allows students to review and assimilate the skills and concepts they learned separately over several sections.
Early Graphing. Chapter 1 connects formulas and mathematical models to data displayed by bar and line graphs. The rectangular coordinate system is introduced in Chapter 3. Graphs appear in nearly every section and Exercise Set. Examples and exercises use graphs to explore relationships between data and to provide ways of visualizing a problem’s solution.
Geometric Problem Solving. Chapter 2 (Linear Equations and Inequalities in One Variable) contains a section that teaches geometric concepts that are important to a student’s understanding of algebra. There is frequent emphasis on problem solving in geometric situations, as well as on geometric models that allow students to visualize algebraic formulas.
Section Objectives. Learning objectives are clearly stated at the beginning of each section. These objectives help students recognize and focus on the section’s most important ideas. The objectives are restated in the margin at their point of use.
Integration of Technology Using Graphic and Numerical Approaches to Problems. Side-by-side features in the technology boxes connect algebraic solutions to graphic and numerical approaches to problems. Although the use of graphing utilities is optional, students can use the explanatory voice balloons to understand different approaches to problems even if they are not using a graphing utility in the course.
Chapter Review Grids. Each chapter contains a review chart that summarizes the definitions and concepts in every section of the chapter. Examples that illustrate these key concepts are also included in the chart.
End-of-Chapter Materials. A comprehensive collection of Review Exercises for each of the chapter’s sections follows the review grid. This is followed by a Chapter Test that enables students to test their understanding of the material covered in the chapter. Beginning with Chapter 2, each chapter concludes with a comprehensive collection of mixed Cumulative Review Exercises.
Blitzer Bonuses. These enrichment essays provide historical, interdisciplinary, and otherwise interesting connections to the algebra under study, showing students that math is an interesting and dynamic discipline.
Discovery. Discover for Yourself boxes, found throughout the text, encourage students to further explore algebraic concepts. These explorations are optional and their omission does not interfere with the continuity of the topic under consideration.
Chapter Projects. At the end of each chapter is a collaborative activity that gives students the opportunity to work cooperatively as they think and talk about mathematics. Additional group projects can be found in the Instructor’s Resource Manual. Many of these exercises should result in interesting group discussions.
I hope that my passion for teaching, as well as my respect for the diversity of students I have taught and learned from over the years, is apparent throughout this new edition. By connecting algebra to the whole spectrum of learning, it is my intent to show students that their world is profoundly mathematical, and indeed, p is in the sky.
Robert Blitzer
Resources for the Fourth Edition For Students
Student Solutions Manual. Fully-worked solutions to the odd-numbered section exercises plus all Check Points, Review/Preview Exercises, Mid-Chapter Check Points, Chapter Reviews, Chapter Tests, and Cumulative Reviews.
NEW Learning Guide, organized by the textbook’s learning objectives, helps students learn how to make the most of their textbook and all of the learning tools, including MyMathLab, while also providing additional practice for each section and guidance for test preparation. Published in an unbound, binder-ready format, the Learning Guide can serve as the foundation for a course notebook for students. NEW Lecture Series on DVD has been completely revised to provide students with extra help for each section of the textbook. The Lecture Series on DVD includes Interactive Lectures and Chapter Test Prep videos.
For Instructors
Annotated Instructor’s Edition. Answers to exercises printed on the same text page with graphing answers in a special Graphing Answer Section in the back of the text. Instructor’s Solutions Manuals. This manual contains fully-worked solutions to the even-numbered section exercises plus all Check Points, Review/Preview Exercises, Mid-Chapter Check Points, Chapter Reviews, Chapter Tests, and Cumulative Reviews. Available in MyMathLab® and on the Instructor’s Resource Center. Instructor’s Resources Manual with Tests. Includes a Mini-Lecture, Skill Builder, and Additional Exercises for every section of the text; two short group Activities per chapter, several chapter test forms, both free-response and multiple-choice, as well as cumulative tests and final exams. Answers to all items also included. Available in MyMath Lab® and on the Instructor’s Resource Center. MyMathLab® Online Course (access code required). MyMathLab® delivers proven results in helping individual students succeed. It provides engaging experiences that personalize, stimulate, and measure learning for each student. And it comes from a trusted partner with educational expertise and an eye on the future.
MyMathLab® Ready to Go Course (access code required). These new Ready to Go courses provide students with all the same great MyMathLab® features that you’re used to, but make it easier for instructors to get started. Each course includes preassigned homework and quizzes to make creating your course even simpler. Ask your Pearson representative about the details for this particular course or to see a copy of this course. To learn more about how MyMathLab® combines proven learning applications with powerful assessment, visit www.mymathlab.com or contact your Pearson representative.
MathXL® Online Courses (access code required) is the homework and assessment engine that runs MyMathLab. (MyMathLab is MathXL plus a learning management system.)
MathXL is available to qualified adopters. For more information, visit our Web site at www.mathxl.com, or contact your Pearson representative.
TestGen® (www.pearsoned.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing instructors to create multiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. The software and testbank are available for download from Pearson Education’s online catalog.
PowerPoint Lecture Slides (download only). Available through www. pearsonhighered.com or inside your MyMathLab® course, these fully editable lecture slides include definitions, key concepts, and examples for use in a lecture setting and are available for each section of the text.
Acknowledgments
An enormous benefit of authoring a successful series is the broad-based feedback I receive from the students, dedicated users, and reviewers. Every change to this edition is the result of their thoughtful comments and suggestions. I would like to express my appreciation to all the reviewers, whose collective insights form the backbone of this revision. In particular, I would like to thank the following people for reviewing Introductory and Intermediate Algebra for College Students.
Gwen P. Aldridge, Northwest Mississippi Community College
Howard Anderson, Skagit Valley College
John Anderson, Illinois Valley Community College
Michael H. Andreoli, Miami Dade College—North Campus
Jan Archibald, Ventura College
Jana Barnard, Angelo State University
Donna Beatty, Ventura College
Michael S. Bowen, Ventura College
Gale Brewer, Amarillo College
Carmen Buhler, Minneapolis Community & Technical College
Hien Bui, Hillsborough Community College
Warren J. Burch, Brevard Community College
Alice Burstein, Middlesex Community College
Edie Carter, Amarillo College
Thomas B. Clark, Trident Technical College
Sandra Pryor Clarkson, Hunter College
Sally Copeland, Johnson County Community College
Carol Curtis, Fresno City College
Bettyann Daley, University of Delaware
Robert A. Davies, Cuyahoga Community College
Paige Davis, Lurleen B. Wallace Community College
Jill DeWitt, Baker College
Ben Divers, Jr., Ferrum College
Irene Doo, Austin Community College
Charles C. Edgar, Onondaga Community College
Karen Edwards, Diablo Valley College
Rhoderick Fleming, Wake Technical Community College
Susan Forman, Bronx Community College
Wendy Fresh, Portland Community College
Donna Gerken, Miami Dade College
Marion K. Glasby, Anne Arundel Community College
Sue Glascoe, Mesa Community College
Gary Glaze, Eastern Washington University
Jay Graening, University of Arkansas
Christina M. Gundlach, Eastern Connecticut State University
Robert B. Hafer, Brevard Community College
Mary Lou Hammond, Spokane Community College
Andrea Hendricks, Georgia Perimeter College
Donald Herrick, Northern Illinois University
Beth Hooper, Golden West College
Sandee House, Georgia Perimeter College
Tracy Hoy, College of Lake County
Laura Hoye, Trident Community College
Margaret Huddleston, Schreiner University
Marcella Jones, Minneapolis Community and Technical College
Shelbra B, Jones, Wake Technical Community College
Judy Kasabian, Lansing Community College
Sharon Keenee, Georgia Perimeter College
Gary Kersting, North Central Michigan College
Dennis Kimzey, Rogue Community College
Kandace Kling, Portland Community College
Gary Knippenberg, Lansing Community College
Mary A. Koehler, Cuyahoga Community College
Kristi Laird, Jackson Community College
Scot Leavitt, Portland Community College
Robert Leibman, Austin Community College
Jennifer Lempke, North Central Michigan College
Sandy Lofstock, St. Petersburg College
Ann M. Loving, J. Sargent Reynolds Community College
Kent MacDougall, Temple College
Hank Martel, Broward Community College
Diana Martelly, Miami Dade College
Kim Martin, Southeastern Illinois College
John Robert Martin, Tarrant County College
Mikal McDowell, Cedar Valley College
Lisa McMillen, Baker College
Irwin Metviner, State University of New York at Old Westbury
Jean P. Millen, Georgia Perimeter College
Lawrence Morales, Seattle Central Community College
Terri Moser, Austin Community College
Robert Musselman, California State University, Fresno
Kamilia Nemri, Spokane Community College
Allen R. Newhart, Parkersburg Community College
Lois Jean Nieme, Minneapolis Community and Technical College
Steve O’Donnell, Rogue Community College
Peg Pankowski, Community College of Allegheny County—South Campus
Jeff Parent, Oakland Community College
Robert Patenade, College of the Canyons
Jill Rafael, Sierra College
James Razavi, Sierra College
Christopher Reisch, The State University of New York at Buffalo
Tian Ren, Queensborough Community College
Nancy Ressler, Oakton Community College
Katalin Rozsa, Mesa Community College
Haazim Sabree, Georgia Perimeter College
Scott W. Satake, Eastern Washington University
Mike Schramm, Indian River Community College
Chris Schultz, Iowa State University
Shannon Schumann, University of Phoenix
Terri Seiver, San Jacinto College
Kathy Shepard, Monroe County Community College
Gayle Smith, Lane Community College
Linda Smoke, Central Michigan University
Dick Spangler, Tacoma Community College
Janette Summers, University of Arkansas
Kory Swart, Kirkwood Community College
Robert Thornton, Loyola University
Lucy C. Thrower, Francis Marion College
Richard Townsend, North Carolina Central University
Andrew Walker, North Seattle Community College
Kim Ward, Eastern Connecticut State University
Kathryn Wetzel, Amarillo College
Margaret Williamson, Milwaukee Area Technical College
Peggy Wright, Cuesta College
Roberta Yellott, McNeese State University
Marilyn Zopp, McHenry County College
Additional acknowledgments are extended to Dan Miller and Kelly Barber, for preparing the solutions manuals and the new Learning Guide, Brad Davis, for preparing the answer section and serving as accuracy checker, the Preparé, Inc. formatting team, for the book’s brilliant paging, Brian Morris at Scientific Illustrators, for superbly illustrating the book, Sheila Norman, photo researcher, for obtaining the book’s new photographs, and Rebecca Dunn, project manager, and Kathleen Manley, production editor, whose collective talents kept every aspect of this complex project moving through its many stages.
I would like to thank my editors at Pearson, Paul Murphy and Dawn Giovanniello, who with the assistance of Chelsea Pingree guided and coordinated the book from manuscript through production. Thanks to Beth Paquin and Nancy Goulet for the cover and Ellen Pettengell for the interior design. Finally, thanks to the entire Pearson sales force for your confidence and enthusiasm about the book.
Robert Blitzer
This page intentionally left blank
To the Student
The bar graph shows some of the qualities that students say make a great teacher.
Qualities That Make a Great Teacher
It was my goal to incorporate each of the qualities that make a great teacher throughout the pages of this book.
Explains Things Clearly
I understand that your primary purpose in reading Introductory andIntermediate Algebra for College Students is to acquire a solid understanding of the required topics in your algebra course. In order to achieve this goal, I’ve carefully explained each topic. Important definitions and procedures are set off in boxes, and worked-out examples that present solutions in a step-by-step manner appear in every section. Each example is followed by a similar matched problem, called a Check Point, for you to try so that you can actively participate in the learning process as you read the book. (Answers to all Check Points appear in the back of the book.)
Funny/Entertaining
Who says that an algebra textbook can’t be entertaining? From our quirky cover to the photos in the chapter and section openers, prepare to expect the unexpected. I hope some of the book’s enrichment essays, called Blitzer Bonuses, will put a smile on your face from time to time.
Helpful
I designed the book’s features to help you acquire knowledge of introductory andintermediatealgebra, as well as to show you how algebra can solve authentic problems that apply to your life. These helpful features include:
Explanatory Voice Balloons: Voice balloons are used in a variety of ways to make math less intimidating. They translate algebraic language into everyday English, help clarify problem-solving procedures, present alternative ways of understanding concepts, and connect new concepts to concepts you have already learned.
Great Question!: The book’s Great Question! boxes are based on questions students ask in class. The answers to these questions give suggestions for problem solving, point out common errors to avoid, and provide informal hints and suggestions.
Achieving Success: The book’s Achieving Success boxes give you helpful strategies for success in learning algebra, as well as suggestions that can be applied for achieving your full academic potential in future college coursework.
Detailed Chapter Review Charts: Each chapter contains a review chart that summarizes the definitions and concepts in every section of the chapter. Examples that illustrate these key concepts are also included in the chart. Review these summaries and you’ll know the most important material in the chapter!
Passionate about Their Subject
I passionately believe that no other discipline comes close to math in offering a more extensive set of tools for application and development of your mind. I wrote the book in Point Reyes National Seashore, 40 miles north of San Francisco. The park consists of 75,000 acres with miles of pristine surf-washed beaches, forested ridges, and bays bordered by white cliffs. It was my hope to convey the beauty and excitement of mathematics using nature’s unspoiled beauty as a source of inspiration and creativity. Enjoy the pages that follow as you empower yourself with the algebra needed to succeed in college, your career, and your life.
Regards
Bob
Robert Blitzer
About the Author
Bob Blitzer is a native of Manhattan andreceived a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University. Bob’s love for teaching mathematics was nourished for nearly 30 years at Miami Dade College, where he received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College and an endowed chair based on excellence in the classroom. In addition to Introductory andIntermediate Algebra for College Students, Bob has written textbookscovering introductory algebra, intermediate algebra, college algebra, algebra and trigonometry, precalculus, and liberal arts mathematics, all published by Pearson. When not secluded in his Northern California writer’s cabin, Bob can be found hiking the beaches and trails of Point Reyes National Seashore, and tending to the chores required by his beloved entourage of horses, chickens, and irritable roosters.
Bob and his horse Jerid
This page intentionally left blank
Variables, Real Numbers, and Mathematical Models
What can algebra possibly tell me about
the rising cost of movie ticket prices over the years? how I can stretch or shrink my life span? the widening imbalance between numbers of women and men on college campuses? the number of calories I need to maintain energy balance? how much I can expect to earn at my first job after college?
In this chapter, you will learn how the special language of algebra describes your world.
Here’s where you will find these applications:
Movie ticket prices: Exercise Set 1.1, Exercises 83–84
Stretching or shrinking my life span: Exercise Set 1.6, Exercises 101–110
College gender imbalance: Section 1.7, Example 9
Caloric needs: Section 1.8, Example 12; Exercise Set 1.8, Exercises 97–98
Anticipated earnings at my first job after college: Exercise Set 1.8, Exercises 99–100.
Objectives
1 Evaluate algebraic expressions.
2 Translate English phrases into algebraic expressions.
3 Determine whether a number is a solution of an equation.
4 Translate English sentences into algebraic equations.
5 Evaluate formulas.
Introduction to Algebra: Variables and Mathematical Models
You are thinking about buying a highdefinition television. How much distance should you allow between you and the TV for pixels to be undetectable and the image to appear smooth?
Algebraic Expressions
Let’s see what the distance between you and your TV has to do with algebra. The biggest difference between arithmetic and algebra is the use of variables in algebra. A variable is a letter that represents a variety of different numbers. For example, we can let x represent the diagonal length, in inches, of a high-definition television. The industry rule for most of the current HDTVs on the market is to multiply this diagonal length by 2.5 to get the distance, in inches, at which a person with perfect vision can see a smooth image. This can be written 2.5 x, but it is usually expressed as 2.5x. Placing a number and a letter next to one another indicates multiplication.
Notice that 2.5x combines the number 2.5 and the variable x using the operation of multiplication. A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Here are some examples of algebraic expressions:
1 Evaluate algebraic expressions.
Great Question!
Are variables always represented by x?
No. As you progress through algebra, you will often see x, y, and z used, but any letter can be used to represent a variable. For example, if we use l to represent a TV’s diagonal length, your ideal distance from the screen is described by the algebraic expression 2.5l
Evaluating Algebraic Expressions
We can replace a variable that appears in an algebraic expression by a number. We are substituting the number for the variable. The process is called evaluating the expression. For example, we can evaluate 2.5x (the ideal distance between you and your x-inch TV) for x 50. We substitute 50 for x. We obtain 2.5 50, or 125. This means that if the diagonal length of your TV is 50 inches, your distance from the screen should be 125inches. Because 12 inches 1 foot, this distance is 125 12 feet, or approximately 10.4 feet.
Many algebraic expressions involve more than one operation. The order in which we add, subtract, multiply, and divide is important. In Section 1.8, we will discuss the rules for the order in which operations should be done. For now, follow this order:
Great Question!
What am I supposed to do with the worked examples?
Study the step-by-step solutions in these examples. Reading the solutions slowly and with great care will prepare you for success with the exercises in the Exercise Sets.
A First Look at Order of Operations
1. Perform all operations within grouping symbols, such as parentheses.
2. Do all multiplications in the order in which they occur from left to right.
3. Do all additions and subtractions in the order in which they occur from left to right.
EXAMPLE 1 Evaluating Expressions
Evaluate each algebraic expression for x 5: a. 3 4x b. 4(x 3).
Solution
a. We begin by substituting 5 for x in 3 4x. Then we follow the order of operations: Multiply first, and then add.
3+4x =3+4 5
Replace x with 5. 3 20 Perform the multiplication: 4 5 20. 23 Perform the addition.
Great Question!
Why is it so important to work each of the book’s CheckPoints?
You learn best by doing. Do not simply look at the worked examples and conclude that you know how to solve them. To be sure you understand the worked examples, try each Check Point. Check your answer in the answer section before continuing your reading. Expect to read this book with pencil and paper handy to work the Check Points.
b. We begin by substituting 5 for x in 4(x 3). Then we follow the order of operations: Perform the addition in parentheses first, and then multiply.
4(x+3)
=4(5+3)
Replace x with 5. 4(8) Perform the addition inside the parentheses: 5 3 8. 4(8) can also be written as 4 8. 32 Multiply.
CHECK POINT 1
Evaluate each expression for x 10: a. 6 2x b. 2(x 6).
Example 2 illustrates that algebraic expressions can contain more than one variable.
EXAMPLE 2 Evaluating Expressions
Evaluate each algebraic expression for x 6 and y 4:
a. 5x 3y b. 3x 5y 2 2x y
2 Translate English phrases into algebraic expressions.
Solution
a.
Replace x with 6.
This is the given algebraic expression. 5x-3y =5 6-3 4
30 12
Replace y with 4.
We are evaluating the expression for x 6 and y 4.
Multiply:5 6 30 and 3 4 12. 18 Subtract.
b. This is the given algebraic expression.
Replace x with 6.
3 6+5 4+2 2 6-4 = 3x+5y+2 2x-y
Replace y with 4.
We are evaluating the expression for x 6 and y 4. 18 20 2 12 4
Multiply: 3 6 18,5 4 20, and 2 6 12. 40 8
Add in the numerator.
Subtract in the denominator.
5 Simplify by dividing 40 by 8.
CHECK POINT 2 Evaluate each algebraic expression for x 3 and y 8:
Translating to Algebraic Expressions
Problem solving in algebra often requires the ability to translate word phrases into algebraic expressions. Table 1.1 lists some key words associated with the operations of addition, subtraction, multiplication, and division.
Table 1.1
Key Words for Addition, Subtraction, Multiplication, and Division
Key words plusminustimesdivide sumdifferenceproductquotient more thanless thantwiceratio increased bydecreased bymultiplied bydivided by
EXAMPLE 3 Translating English Phrases into Algebraic Expressions
Write each English phrase as an algebraic expression. Let the variable x represent the number.
a. the sum of a number and 7
b. ten less than a number
c. twice a number, decreased by 6
d. the product of 8 and a number
e. three more than the quotient of a number and 11
Solution
a.
The sum ofa number and 7
x+7
The algebraic expression for “the sum of a number and 7” is x 7.
b. x-10
Ten less thana number
The algebraic expression for “ten less than a number” is x 10.
c.
d.
e.
Twice a number,decreased by 6
2x-6
The algebraic expression for “twice a number, decreased by 6” is 2x 6.
The product of 8 and a number
8 x
The algebraic expression for “the product of 8 and a number” is 8 x, or 8x.
Three more thanthe quotient of a number and 11 x 11
+3
The algebraic expression for “three more than the quotient of a number and 11” is x 11 3.
Great Question!
Are there any helpful hints I can use when writing algebraic expressions forEnglish phrases involving subtraction?
Yes. Pay close attention to order when translating phrases involving subtraction.
PhraseTranslation
A number decreased by 7 x 7
A number subtracted from 7 7 x
Seven less than a number x 7
Seven less a number7 x
Think carefully about what is expressed in English before you translate into the language of algebra.
CHECK POINT 3
Write each English phrase as an algebraic expression. Let the variable x represent the number.
a. the product of 6 and a number
b. a number added to 4
c. three times a number, increased by 5
d. twice a number subtracted from 12
e. the quotient of 15 and a number
3 Determine whether a number is a solution of an equation.
Equations
An equation is a statement that two algebraic expressions are equal. An equation always contains the equality symbol . Here are some examples of equations:
6x 16 464y 2 2y 62(z 1) 5(z 2).
Solutions of an equation are values of the variable that make the equation a true statement. To determine whether a number is a solution, substitute that number for the variable and evaluate each side of the equation. If the values on both sides of the equation are the same, the number is a solution.
EXAMPLE 4
Determining Whether Numbers Are Solutions of Equations
Determine whether the given number is a solution of the equation.
a. 6x 16 46; 5
Solution
a.
6x+16=46
Is 5 a solution?
6 5+16 46
30 16 46
46=46 This statement is true.
b. 2(z 1) 5(z 2); 7
This is the given equation.
b.
To determine whether 5 is a solution, substitute 5 for x. The question mark over the equal sign indicates that we do not yet know if the statement is true.
Multiply: 6 5 30.
Add:30 16 46.
Because the values on both sides of the equation are the same, the number 5 is a solution of the equation.
2(z+1)=5(z-2)
Is 7 a solution?
2(7+1) 5(7-2)
2(8) 5(5)
16=25 This statement is false.
This is the given equation.
To determine whether 7 is a solution, substitute 7 for z.
Perform operations in parentheses: 7 1 8 and 7 2 5.
Multiply:2 8 16 and 5 5 25.
Because the values on both sides of the equation are not the same, the number 7 is not a solution of the equation.
CHECK POINT 4 Determine whether the given number is a solution of the equation.
a. 9x 3 42; 6
b. 2(y 3) 5y 3; 3
4 Translate English sentences into algebraic equations.
Translating to Equations
Earlier in the section, we translated English phrases into algebraic expressions. Now we will translate English sentences into equations. You’ll find that there are a number of different words and phrases for an equation’s equality symbol. equals gives yields is the same as is/was/will be
EXAMPLE 5 Translating English Sentences into Algebraic Equations
Write each sentence as an equation. Let the variable x represent the number.
a. The product of 6 and a number is 30.
b. Seven less than 3 times a number gives 17.
Solution
a. 6x=
Great Question!
I don’t know how to find the solutions of the equations in Example 5. ShouldI be worried?
Do not be concerned. We’ll discuss how to solve these equations in Chapter 2.
These are meant to be amusing.
Algebraically, this is the important item.
5 Evaluate formulas.
The product of 6 and a number is 30 30.
The equation for “the product of 6 and a number is 30” is 6x 30.
Seven less than 3 times a number gives 17.
b. 3x-7=17
The equation for “seven less than 3 times a number gives 17” is 3x 7 17.
CHECK POINT 5
Write each sentence as an equation. Let the variable x represent the number.
a. The quotient of a number and 6 is 5.
b. Seven decreased by twice a number yields 1.
Great Question!
When translating English sentences containing commas into algebraic equations, should I pay attention to the commas?
Commas make a difference. In English, sentences and phrases can take on different meanings depending on the way words are grouped with commas. Some examples:
Woman, without her man, is nothing
Woman, without her, man is nothing
Do not break your bread or roll in your soup.
Do not break your bread, or roll in your soup.
The product of 6 and a number increased by 5 is 30: 6( 5) 30
The product of 6 and a number, increased by 5, is 30: 6 5 30
Formulas and Mathematical Models
One aim of algebra is to provide a compact, symbolic description of the world. These descriptions involve the use of formulas. A formula is an equation that expresses a relationship between two or more variables. For example, one variety of crickets chirps
Source: B. E. Pruitt et al., Human Sexuality,
faster as the temperature rises. You can calculate the temperature by counting the number of times a cricket chirps per minute and applying the following formula:
T 0.3n 40.
In the formula, T is the temperature, in degrees Fahrenheit, and n is the number of cricket chirps per minute. We can use this formula to determine the temperature if you are sitting on your porch and count 80 chirps per minute. Here is how to do so:
T 0.3n 40
T 0.3(80) 40
T 24 40
T 64.
This is the given formula.
Substitute 80 for n
Multiply:0.3(80) 24.
Add.
When there are 80 cricket chirps per minute, the temperature is 64 degrees.
The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models. We often say that these formulas model, or describe, the relationship among the variables.
In creating mathematical models, we strive for both accuracy and simplicity. For example, the formula T 0.3n 40 is relatively simple to use. However, you should not get upset if you count 80 cricket chirps and the actual temperature is 62 degrees, rather than 64 degrees, as predicted by the formula. Many mathematical models give an approximate, rather than an exact, description of the relationship between variables.
Sometimes a mathematical model gives an estimate that is not a good approximation or is extended to include values of the variable that do not make sense. In these cases, we say that model breakdown has occurred. Here is an example:
Use the mathematical model = 0.3 + 40 with = 1200 (1200 cricket chirps per minute).
T=0.3(1200)+40=360+40=400
At 400º F, forget about 1200 chirps per minute! At this temperature, the cricket would "cook" and, alas, all chirping would cease.
EXAMPLE 6 Age at Marriage and the Probability of Divorce
Divorce rates are considerably higher for couples who marry in their teens. The line graphs in Figure 1.1 show the percentages of marriages ending in divorce based on the wife’s age at marriage.
2007 Probability of Divorce, by Wife’s Age at Marriage
Figure 1.1
Prentice Hall,
Here are two mathematical models that approximate the data displayed by the line graphs:
Wife is under 18 at time of marriage.
d=4n+5
Wife is over 25 at time of marriage.
d=2.3n+1.5.
In each model, the variable n is the number of years after marriage and the variable d is the percentage of marriages ending in divorce.
a. Use the appropriate formula to determine the percentage of marriages ending in divorce after 10 years when the wife is over 25 at the time of marriage.
b. Use the appropriate line graph in Figure 1.1 to determine the percentage of marriages ending in divorce after 10 years when the wife is over 25 at the time of marriage.
c. Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 10 years as shown by the graph? By how much?
Solution
a. Because the wife is over 25 at the time of marriage, we use the formula on the right, d 2.3n 1.5. To find the percentage of marriages ending in divorce after 10 years, we substitute 10 for n and evaluate the formula.
d 2.3n 1.5 This is one of the two given mathematical models.
d 2.3(10) 1.5 Replace n with 10.
d 23 1.5
Multiply:2.3(10) 23.
d 24.5 Add.
The model indicates that 24.5% of marriages end in divorce after 10 years when the wife is over 25 at the time of marriage.
b. Now let’s use the line graph that shows the percentage of marriages ending in divorce when the wife is over 25 at the time of marriage. The graph is shown again in Figure 1.2. To find the percentage of marriages ending in divorce after 10 years:
Probability of Divorce When Wife Is over 25 at Marriage
This is the point of interest.
The actual data displayed by the graph indicate that 25% of these marriages end in divorce after 10 years.
Figure 1.2
Another random document with no related content on Scribd:
The Project Gutenberg eBook of Among the gnomes
This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Among the gnomes
An occult tale of adventure in the Untersberg
Author: Franz Hartmann
Illustrator: S. Hartmann Sedgwick
Release date: June 18, 2024 [eBook #73860]
Language: English
Original publication: London: T. Fisher Unwin, 1895
Credits: Susan E., David E. Brown, and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK AMONG THE GNOMES ***
