finite mathematics and calculus with applications

Preface ixDear Student xxiiPrerequisite Skills Diagnostic Test xxiiiPolynomials R-2Factoring R-5Rational Expressions R-8Equations R-11 Inequalities R-16Exponents R-21Radicals R-25 Slopes

Finite Mathematics and Calculus with Applications

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Library of Congress Cataloging-in-Publication Data

Lial, Margaret L

Finite mathematics and calculus with applications — 9th

ed / Margaret L Lial, Raymond N Greenwell, Nathan P

1 Mathematics — Textbooks 2 Calculus— Textbooks I

Greenwell, Raymond N II Ritchey, Nathan P III Title

QA37.3.L54 2013

510 — dc22

2010031432Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc All rights reserved No part of this publication may be reproduced,

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1 2 3 4 5 6 7 8 9 10—QG—15 14 13 12 11

ISBN-10: 0-321-74908-1ISBN-13: 978-0-321-74908-6

N O T I C E : This work is protected by U.S copyright laws and

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www.pearsonhighered.com

Preface ixDear Student xxiiPrerequisite Skills Diagnostic Test xxiii

Polynomials R-2Factoring R-5Rational Expressions R-8Equations R-11

Inequalities R-16Exponents R-21Radicals R-25

Slopes and Equations of Lines 2Linear Functions and Applications 17The Least Squares Line 25

CHAPTER 1 REVIEW 38EXTENDED APPLICATION Using Extrapolation to Predict Life Expectancy 42

Solution of Linear Systems by the Echelon Method 45Solution of Linear Systems by the Gauss-Jordan Method 54Addition and Subtraction of Matrices 70

Multiplication of Matrices 77Matrix Inverses 87

Input-Output Models 97CHAPTER 2 REVIEW 104EXTENDED APPLICATION Contagion 110

Graphing Linear Inequalities 113Solving Linear Programming Problems Graphically 120Applications of Linear Programming 126

CHAPTER 3 REVIEW 134EXTENDED APPLICATION Sensitivity Analysis 137

3.3 3.2 3.1

2.6 2.5 2.4 2.3 2.2 2.1

1.3 1.2 1.1

R.7 R.6 R.5 R.4 R.3 R.2 R.1

Linear Programming:The Simplex Method 142Slack Variables and the Pivot 143

Maximization Problems 150Minimization Problems; Duality 161Nonstandard Problems 170CHAPTER 4 REVIEW 179EXTENDED APPLICATION Using Integer Programming in the Stock-Cutting Problem 183

Simple and Compound Interest 188Future Value of an Annuity 200Present Value of an Annuity; Amortization 209CHAPTER 5 REVIEW 218

EXTENDED APPLICATION Time, Money, and Polynomials 222

Statements 225Truth Tables and Equivalent Statements 233The Conditional and Circuits 240

More on the Conditional 250Analyzing Arguments and Proofs 257Analyzing Arguments with Quantifiers 266CHAPTER 6 REVIEW 274

EXTENDED APPLICATION Logic Puzzles 279

Applications of Venn Diagrams 292Introduction to Probability 302Basic Concepts of Probability 311Conditional Probability; Independent Events 322Bayes’ Theorem 336

CHAPTER 7 REVIEW 343EXTENDED APPLICATION Medical Diagnosis 350

7.6 7.5 7.4 7.3 7.2 7.1

CHAPTER

6.6 6.5 6.4 6.3 6.2 6.1

CHAPTER

5.3 5.2 5.1

CHAPTER

4.4 4.3 4.2 4.1

Counting Principles; Further Probability Topics 352

The Multiplication Principle; Permutations 353Combinations 361

Probability Applications of Counting Principles 370Binomial Probability 381

Probability Distributions; Expected Value 389CHAPTER 8 REVIEW 400

EXTENDED APPLICATION Optimal Inventory for a Service Truck 405

EXTENDED APPLICATION Statistics in the Law—The Castaneda Decision 449

Nonlinear Functions 452

Properties of Functions 453Quadratic Functions;Translation and Reflection 465Polynomial and Rational Functions 475

Exponential Functions 487Logarithmic Functions 497Applications: Growth and Decay; Mathematics of Finance 510CHAPTER 10 REVIEW 518

EXTENDED APPLICATION Power Functions 526

The Derivative 529

Limits 530Continuity 548Rates of Change 557Definition of the Derivative 570Graphical Differentiation 588CHAPTER 11 REVIEW 594EXTENDED APPLICATION A Model for Drugs Administered Intravenously 601

Calculating the Derivative 604Techniques for Finding Derivatives 605Derivatives of Products and Quotients 619The Chain Rule 626

Derivatives of Exponential Functions 636Derivatives of Logarithmic Functions 644CHAPTER 12 REVIEW 651

EXTENDED APPLICATION Electric Potential and Electric Field 656

Graphs and the Derivative 659Increasing and Decreasing Functions 660Relative Extrema 671

Higher Derivatives, Concavity, and the Second Derivative Test 682Curve Sketching 695

CHAPTER 13 REVIEW 704EXTENDED APPLICATION A Drug Concentration Model for Orally Administered Medications 708

Applications of the Derivative 711Absolute Extrema 712

Applications of Extrema 721Further Business Applications: Economic Lot Size; Economic Order Quantity;Elasticity of Demand 730

Implicit Differentiation 739Related Rates 744Differentials: Linear Approximation 751CHAPTER 14 REVIEW 757

EXTENDED APPLICATION A Total Cost Model for a Training Program 761

Integration 763Antiderivatives 764Substitution 776Area and the Definite Integral 784The Fundamental Theorem of Calculus 796The Area Between Two Curves 806Numerical Integration 816

CHAPTER 15 REVIEW 824EXTENDED APPLICATION Estimating Depletion Dates for Minerals 829

15.6 15.5 15.4 15.3 15.2 15.1

CHAPTER

14.6 14.5 14.4 14.3 14.2 14.1

CHAPTER

13.4 13.3 13.2 13.1

CHAPTER

12.5 12.4 12.3 12.2 12.1

CHAPTER

13

14

15 12

Further Techniques and Applications of Integration 833Integration by Parts 834

Volume and Average Value 842Continuous Money Flow 849Improper Integrals 856Solution of Elementary and Separable Differential Equations 862CHAPTER 16 REVIEW 875

EXTENDED APPLICATION Estimating Learning Curves in Manufacturing with Integrals 880

CHAPTER 17 REVIEW 939EXTENDED APPLICATION Using Multivariable Fitting to Create a Response Surface Design 945

Probability and Calculus 949

Continuous Probability Models 950Expected Value and Variance of Continuous Random Variables 961Special Probability Density Functions 970

CHAPTER 18 REVIEW 982EXTENDED APPLICATION Exponential Waiting Times 987

1 Formulas from Geometry

2 Area Under a Normal Curve

Answers to Selected Exercises A-17 Credits C-1

Index of Applications I-1Index I-7

Review Exercises

Finite Mathematics and Calculus with Applications is a thorough, application-oriented text for

students majoring in business, management, economics, or the life or social sciences In addition to its clear exposition, this text consistently connects the mathematics to career and everyday-life situations A prerequisite of two years of high school algebra is assumed A renewed focus on quick and effective assessments, new applications and exercises, as well as other new learning tools make this 9th edition an even richer learning resource for students.

Our Approach

Our main goal is to present finite mathematics and applied calculus in a concise and meaningful way so that students can understand the full picture of the concepts they are learning and apply it to real-life situations This is done through a variety of ways.

Focus on Applications Making this course meaningful to students is critical to their success Applications of the mathematics are integrated throughout the text in the exposition,

the examples, the exercise sets, and the supplementary resources Finite Mathematics and

Calculus with Applications presents students with a myriad of opportunities to relate what

they’re learning to career situations through the Apply It questions, the applied examples, and the Extended Applications To get a sense of the breadth of applications presented, look at

the Index of Applications in the back of the book or the extended list of sources of real-world data on www.pearsonhighered.com/mathstatsresources.

Pedagogy to Support Students Students need careful explanations of the mathematics along with examples presented in a clear and consistent manner Additionally students and instructors should have a means to assess the basic prerequisite skills This can now be done

with the Prerequisite Skills Diagnostic Test located just before Chapter R In addition, the

stu-dents need a mechanism to check their understanding as they go and resources to help them

remediate if necessary Finite Mathematics and Calculus with Applications has this support built into the pedagogy of the text through fully developed and annotated examples, Your

Turn exercises, For Review references, and supplementary material.

Beyond the Textbook Students today take advantage of a variety of resources and delivery

methods for instruction As such, we have developed a robust MyMathLab course for Finite

Mathematics and Calculus with Applications MyMathLab has a established and

well-documented track record of helping students succeed in mathematics The MyMathLab

online course for Finite Mathematics and Calculus with Applications contains over 6700

exer-cises to challenge students and provides help when they need it Students who learn best by seeing and hearing can view section- and example-level videos within MyMathLab or on the book-specific DVD-Rom These and other resources are available to students as a unified and reliable tool for their success.

New to the Ninth Edition

Based on the authors’ experience in the classroom along with feedback from many instructors across the country, the focus of this revision is to improve the clarity of the presentation and provide students with more opportunities to learn, practice, and apply what they’ve learned on their own This is done in both the presentation of the content and

in new features added to the text.

Preface

ix

New and Revised Content

• Chapter R The flow of the material was improved by reordering some exercises and examples Exercises were added to Section R.1 (on performing algebraic operations) and Section R.5 (on solving inequalities).

• Chapter 1 Changes in the presentation were made throughout to increase clarity, ing adding some examples and rewriting others Terminology in Section 1.2 was adjusted

includ-to be more consistent with usage in economics.

• Chapter 2 Section 2.1 was changed so that only systems of two equations are solved by the echelon method, while systems with three or more equations are solved using the Gauss-Jordan method in Section 2.2 The discussion of subtraction of matrices in Section 2.3 was simplified.

• Chapter 3 The concept of bounded and unbounded regions was moved from Section 3.2

to Section 3.1, where such regions are first encountered An Extended Application on

sensi-tivity analysis was added to this chapter.

• Chapter 4 Exercises 25 through 30 in Section 4.1 were modified to clarify the role of slack variables Exercise 30 in Section 4.2 was modified to amplify how multiple solutions may occur The method for handling ties in nonstandard problems in Section 4.4 was improved.

• Chapter 5 In Section 5.1, examples and accompanying exercises were added covering how

to solve for the interest rate and how to find the compounding time, both with a graphing culator and with logarithms The explanation of the rule of 70 and the rule of 72 was im- proved Material on continuous compounding was also added to Section 5.1 In Section 5.3,

cal-an example cal-and accompcal-anying exercises were added on how a local-an ccal-an be paid off early

• Chapter 6 Many exercises in this chapter were revised so that the information would be more relevant to students For example, tax references include scholarships, tuition, paychecks, reporting tips, filing taxes, inheritances, and tuition deductions Law references include car accidents, contracts, lawsuits, driver’s licenses, and marriage, and warranty references cover iPhones and eBay In Section 6.5, applications were revised to give more diversity in topics.

• Chapter 7 Empirical probability was moved from Section 7.4 to 7.3 so that methods for determining probability are contained in the same section In Section 7.4, probability distri- butions are emphasized more and a probability distribution example was added The intro- duction to Bayes’ Theorem was rewritten for brevity and clarity in Section 7.6.

• Chapter 8 The notation for combinations was changed from to to be more current and consistent with our notation throughout the book Section 8.3 now includes an example illustrating probabilities using permutations and the multiplication principle.

• Chapter 9 In Section 9.1, a new example was added illustrating a case in which the median is a truer representation of data than the mean

• Chapter 10 The material in Section 10.1 on the Dow Jones Average was updated Material

on even and odd functions was added Material on identifying the degree of a polynomial has been rewritten as an example to better highlight the concept The discussion of the Rule

of 70 and the Rule of 72 was improved A new Extended Application on Power Functions

has been added.

• Chapter 11 In Section 11.1, the introduction of limits was completely revised The ing discussion and example were transformed into a series of examples that progress through different limit scenarios: a function defined at the limit, a function undefined at the limit (a hole in the graph), a function defined at the limit but with a different value than the limit (a piecewise function), and then finally, finding a limit when one does not exist New figures were added to illustrate the different scenarios In Section 11.2 the definition and example of continuity has been revised using a simple process to test for continuity The opening discussion of Section 11.5, showing how to sketch the graph of the derivative given the graph of the original function, was rewritten as an example.

open-C1 n, r 2

a n r b

• Chapter 12 The introduction to the chain rule was rewritten as an example in Section 12.3 Exercise topics were revised to cover subjects such as worldwide Internet users, online learning, and the Gateway Arch.

• Chapter 13 In Section 13.1, the definition of increasing/decreasing functions has been moved to the beginning of the chapter, followed by the discussion of using derivatives to determine where the function increases and decreases The determination of where a func- tion is increasing or decreasing is divided into three examples: when the critical numbers are found by setting the derivative equal to zero, when the critical numbers are found by deter- mining where the derivative is undefined, and when the function has no critical numbers.

• Chapter 14 Changes in the presentation were made throughout to increase clarity and exercise sets were rearranged to improve progression and parity.

• Chapter 15 The social sciences category of exercises was added to Section 15.1, ing the topics of bachelor’s degrees and the number of females earning degrees in dentistry Color was added to the introduction and first example of substitution in Section 15.2 to enable students to follow the substitution more easily

includ-• Chapter 16 In addition to exercises based on real data being updated, examples in this chapter were changed for pedagogical reasons.

• Chapter 17 Graphs generated by Maple™ were added to Examples 2 and 4 in Section 17.3

to assist students in visualizing the concept of relative extrema Material covering utility functions was added to Section 17.4 Many of the figures of three-dimensional surfaces were improved to make them clearer and more attractive.

• Chapter 18 In Section 18.2, an example on how to calculate the probability within one standard deviation of the mean (which is required in many of the exercises) was added The Social Sciences category was added to the exercise set, with exercises on calculating the median, expected value, and standard deviation Topics include the time it takes to learn a task and the age of users of a social network.

Prerequisite Skills Diagnostic Test

The Prerequisite Skills Diagnostic Test gives students and instructors a means to assess the basic prerequisite skills needed to be successful in this course In addition, the answers to the test include references to specific content in Chapter R as applicable so students can zero in

on where they need improvement Solutions to the questions in this test are in Appendix A

More Applications and Exercises

This text is used in large part because of the enormous amounts of real data used in examples and exercises throughout the text This 9th edition will not disappoint in this area We have added or updated more than 20% of the applications and 32% of the examples throughout the text and added or updated more than 600 exercises.

Reference Tables for Exercises

The answers to odd-numbered exercises in the back of the textbook now contain a table referring students to a specific example in the section for help with most exercises For the review exercises, the table refers to the section in the chapter where the topic of that exercise

is first discussed.

Annotated Instructor’s Edition

The annotated instructor’s edition is filled with valuable teaching tips in the margins for those instructors who are new to teaching this course In addition, answers to most exercises are provided directly on the exercise set page along with + symbol next to the most challenging exercises to make assigning and checking homework easier.

PREFACE xi

New to MyMathLab

Available now with Finite Mathematics and Calculus with Applications are the following

resources within MyMathLab that will benefit students in this course.

• “Getting Ready for Finite Mathematics” and “Getting Ready for Applied Calculus” chapters cover basic prerequisite skills

• Personalized Homework allows you to create homework assignments based on the results of student assessments

• Videos with extensive section coverage

• Hundreds more assignable exercises than the previous edition of the text

• Application labels within exercise sets (e.g., “Bus/Econ”) make it easy for you to find types of applications appropriate to your students

• Additional graphing calculator and Excel spreadsheet help

A detailed description of the overall capabilities of MyMathLab is provided

on page xviii

Source Lines

Sources for the exercises are now written in an abbreviated format within the actual exercise

so that students immediately see that the problem comes from, or pulls data from, actual research or industry The complete references are available at www.pearsonhighered.com/ mathstatsresources as well as on page S-1.

Other New Features

We have worked hard to meet the needs of today’s students through this revision In addition

to the new content and resources listed above, there are many new features to this 9th edition

including new and enhanced examples, Your Turn exercises, the inclusion of and instruction for new technology, and new and updated Extended Applications You can

view these new features in context in the following Quick Walk-Through of Finite

Mathematics and Calculus with Applications, 9e.

A Quick Walk-Through of Finite Mathematics and Calculus with Applications, 9e

Mathematics of Finance

5.1 Simple and Compound Interest

5.2 Future Value of an Annuity

5.3 Present Value of an Annuity;Amortization

to amortize a loan.

5

Present Value of an Annuity; Amortization

What monthly payment will pay off a $17,000 car loan in 36 monthly payments at 6% annual interest?

sav-one lump sum today (at the same compound interest rate) in order to produce exactly the ity as follows.

Suppose deposits of R dollars are made at the end of each period for n periods at est rate i per period Then the amount in the account after n periods is the future value of

inter-this annuity:

On the other hand, if P dollars are deposited today at the same compound interest rate i, then at the end of n periods, the amount in the account is If P is the present value

of the annuity, this amount must be the same as the amount S in the formula above; that is,

To solve this equation for P, multiply both sides by

Use the distributive property; also recall that

The amount P is the present value of the annuity The quantity in brackets is abbreviated as

P11 1 i2 n

S 5 R s n 0i 5 Rc11 1 i2in2 1d

FOR REVIEW

Recall from Section R.6 that for

any nonzero number a,

Also, by the product rule for exponents, In par-

ticular, if a is any nonzero number

An Apply It question, typically at the start of a

sec-tion, asks students to consider how to solve a

real-life situation related to the math they are about to

learn The Apply It question is answered in an

application within the section or the exercise set.

(“Apply It” was labeled “Think About It” in the

previous edition.)

䉴

For Review

For Review boxes are provided in the margin as

appropriate, giving students just-in-time help

with skills they should already know but may

have forgotten For Review comments sometimes

include an explanation while others refer students

back to earlier parts of the book for a more thorough

review.

NEW!

Teaching Tips

Teaching Tips are provided in the margins of the

Annotated Instructor’s Edition for those who are

new to teaching this course In addition, answers

to most exercises are provided directly on the

exercise set page making it easier to assign and

check homework.

䉴

NEW!

“Your Turn” Exercises

The Your Turn exercises, following selected

examples, provide students with an easy way to

quickly stop and check their understanding of the

skill or concept being presented Answers are

provided at the end of the section’s exercises.

䉴 䉴

䉴

Caution boxes provide students with a quick

“heads-up” to common student difficulties and

errors.

NEW!

Recognizing New Technology

Material on graphing calculators or Microsoft Excel™ is now set off to make it easier for instructors to use this material or not All of the figures depicting graphing calcu- lator screens have been redrawn to create a more accurate depiction of the math In addition, this edition references and provides students with a transition to the new MathPrint™ operating system of the TI-84 Plus through

the technology notes, a new appendix, and the Graphing

Calculator and Excel Spreadsheet Manual

䉴

Apply It

The solution to the Apply It question often falls in

the body of the text where it can be seen in context

with the mathematics.

䉴

䉴

Exercises

Skill-based problems are followed by

application exercises, which are grouped by

subject with subheads indicating the specific

topic.

Connection exercises integrate topics

pre-sented in different sections or chapters and

are indicated with

Technology exercises are labeled with

for graphing calculator and for spreadsheet.

Writing exercises, labeled with provide

students with an opportunity to explain important mathematical ideas

Exercises that are particularly challenging are

denoted with + in the Annotated Instructor’s Edition only

䉴

1 Explain the difference between the present value of an annuity

and the future value of an annuity For a given annuity, which

is larger? Why?

2 What does it mean to amortize a loan?

Find the present value of each ordinary annuity.

3 Payments of $890 each year for 16 years at 6% compounded

11 $2500; 6% compounded quarterly; 6 quarterly payments

12 $41,000; 8% compounded semiannually; 10 semiannual

payments

13 $90,000; 6% compounded annually; 12 annual payments

14 $140,000; 8% compounded quarterly; 15 quarterly payments

15 $7400; 6.2% compounded semiannually; 18 semiannual payments

16 $5500; 10% compounded monthly; 24 monthly payments Suppose that in the loans described in Exercises 13 –16, the bor- the amount needed to pay off the loan, using either of the two methods described in Example 4.

17 After 3 years in Exercise 13

18 After 5 quarters in Exercise 14

19 After 3 years in Exercise 15

20 After 7 months in Exercise 16 Use the amortization table in Example 5 to answer the ques- tions in Exercises 21–24.

21 How much of the fourth payment is interest?

22 How much of the eleventh payment is used to reduce the debt?

5.3 EXERCISES

23 How much interest is paid in the first 4 months of the loan?

24 How much interest is paid in the last 4 months of the loan?

25 What sum deposited today at 5% compounded annually for 8

years will provide the same amount as $1000 deposited at the end of each year for 8 years at 6% compounded annually?

26 What lump sum deposited today at 8% compounded quarterly

for 10 years will yield the same final amount as deposits of compounded semiannually?

Find the monthly house payments necessary to amortize each loan Then calculate the total payments and the total amount

Business and Economics

35 House Payments Calculate the monthly payment and total amount of interest paid in Example 3 with a 15-year loan, and then compare with the results of Example 3.

36 Installment Buying Stereo Shack sells a stereo system for

$600 down and monthly payments of $30 for the next 3 years If the interest rate is 1.25% per month on the unpaid balance, find

a the cost of the stereo system.

b the total amount of interest paid.

37 Car Payments Hong Le buys a car costing $14,000 He agrees to make payments at the end of each monthly period for

4 years He pays 7% interest, compounded monthly.

a What is the amount of each payment?

b Find the total amount of interest Le will pay.

38 Credit Card Debt Tom Shaffer charged $8430 on his credit card to relocate for his first job When he realized that the monthly, he decided not to charge any more on that account.

5.3 Present Value of an Annuity; Amortization 217

48 Loan Payments When Nancy Hart opened her law office,

she bought $14,000 worth of law books and $7200 worth of

office furniture She paid $1200 down and agreed to amortize

the balance with semiannual payments for 5 years, at 8%

com-pounded semiannually.

a Find the amount of each payment.

b Refer to the text and Figure 13 When her loan had been

reduced below $5000, Nancy received a large tax refund and

this time?

49 House Payments Ian Desrosiers buys a house for $285,000.

He pays $60,000 down and takes out a mortgage at 6.5% on

interest he will pay if the length of the mortgage is

a 15 years;

b 20 years;

c 25 years.

d Refer to the text and Figure 13 When will half the 20-year

loan in part b be paid off?

50 House Payments The Chavara family buys a house for

$225,000 They pay $50,000 down and take out a 30-year

total amount of interest they will pay if the interest rate is

a 6%;

b 6.5%;

c 7%.

d Refer to the text and Figure 13 When will half the 7% loan

in part c be paid off?

51 Refinancing a Mortgage Fifteen years ago, the Budai family

bought a home and financed $150,000 with a 30-year mortgage

at 8.2%.

a Find their monthly payment, the total amount of their

pay-ments, and the total amount of interest they will pay over the

life of this loan.

b The Budais made payments for 15 years Estimate the

unpaid balance using the formula

, and then calculate the total of their remaining payments.

c Suppose interest rates have dropped since the Budai family

took out their original loan One local bank now offers a

30-year mortgage at 6.5% The bank fees for refinancing are

balance of their loan, find their monthly payment Including

ments? Discuss whether or not the family should refinance

with this option.

y 5 R c1 2 1 1 1 i 2i 21n2x2d

d A different bank offers the same 6.5% rate but on a 15-year

mortgage Their fee for financing is $4500 If the Budais pay their monthly payment Including the refinancing fee, what not the family should refinance with this option.

52 Inheritance Deborah Harden has inherited $25,000 from her grandfather’s estate She deposits the money in an account offering 6% interest compounded annually She wants to make (principal and interest) lasts exactly 8 years.

a Find the amount of each withdrawal.

b Find the amount of each withdrawal if the money must last

12 years.

53 Charitable Trust The trustees of a college have accepted a gift of $150,000 The donor has directed the trustees to deposit semiannually The trustees may make equal withdrawals at the end of each 6-month period; the money must last 5 years.

a Find the amount of each withdrawal.

b Find the amount of each withdrawal if the money must last

6 years.

Amortization Prepare an amortization schedule for each loan.

54 A loan of $37,948 with interest at 6.5% compounded annually,

to be paid with equal annual payments over 10 years.

55 A loan of $4836 at 7.25% interest compounded semi-annually,

to be repaid in 5 years in equal semiannual payments.

56 Perpetuity A perpetuity is an annuity in which the payments

go on forever We can derive a formula for the present value of annuity and looking at what happens when n gets larger

and larger Explain why the present value of a perpetuity is given by

57 Perpetuity Using the result of Exercise 56, find the present value of perpetuities for each of the following.

a Payments of $1000 a year with 4% interest compounded

End-of-Chapter Summary provides students

with a quick summary of the key ideas of the

chapter followed by a list of key definitions, terms,

and examples.

䉴

Extended Applications

Extended Applications are provided now at

the end of every chapter as in-depth applied

exercises to help stimulate student interest These activities can be completed individually

or as a group project

CHAPTER 5 Mathematics of Finance

220

15 For a given amount of money at a given interest rate for a given

time period, does simple interest or compound interest produce more interest?

Find the compound amount in each loan.

16 $2800 at 7% compounded annually for 10 years

17 $19,456.11 at 8% compounded semiannually for 7 years

18 $312.45 at 5.6% compounded semiannually for 16 years

19 $57,809.34 at 6% compounded quarterly for 5 years Find the amount of interest earned by each deposit.

20 $3954 at 8% compounded annually for 10 years

21 $12,699.36 at 5% compounded semiannually for 7 years

22 $12,903.45 at 6.4% compounded quarterly for 29 quarters

23 $34,677.23 at 4.8% compounded monthly for 32 months

24 What is meant by the present value of an amount A?

Find the present value of each amount.

25 $42,000 in 7 years, 6% compounded monthly

26 $17,650 in 4 years, 4% compounded quarterly

27 $1347.89 in 3.5 years, 6.77% compounded semiannually

28 $2388.90 in 44 months, 5.93% compounded monthly

29 Write the first five terms of the geometric sequence with a 5 2

43 $11,900 deposited at the beginning of each month for 13 months;

money earns 6% compounded monthly.

44 What is the purpose of a sinking fund?

Find the amount of each payment that must be made into a sinking fund to accumulate each amount

45 $6500; money earns 5% compounded annually for 6 years.

46 $57,000; money earns 4% compounded semiannually for

years.

47 $233,188; money earns 5.2% compounded quarterly for years.

48 $1,056,788; money earns 7.2% compounded monthly for years Find the present value of each ordinary annuity.

49 Deposits of $850 annually for 4 years at 6% compounded

Chapter Review Exercises

Chapter Review Exercises have been slightly

reorganized so that the Concept Check exercises fall within the Chapter Review Exercises This provides students with a more complete review

of both the skills and the concepts they should have mastered in this chapter These exercises in their entirety provide a comprehensive review for

a chapter-level exam.

䉴

TIME, MONEY, AND POLYNOMIALS*

Atime line is often

helpful for evaluating For example, suppose you buy a $1000 CD at time After one year $2500 is added to the CD at By time after another year, your money has grown to $3851 interest, called yield to matu- rity (YTM), did your money

earn? A time line for this ation is shown in Figure 15.

t0.

Assuming interest is compounded annually at a rate i, and using the

compo nd interest form la gi es the follo ing description of the YTM

To determine the yield to maturity, we must solve this equation for

i Since the quantity is repeated, let and first solve the second-degree (quadratic) polynomial equation for x.

We can use the quadratic formula with and

reject the negative value because the final accumulation is greater tive rates may be meaningful By checking in the first equation, we see that the yield to maturity for the CD is 7.67%.

Now let us consider a more complex but realistic problem pose Austin Caperton has contributed for 4 years to a retirement fund.

Sup-ning of the next 3 years, he contributed $5840, $4000, and $5200, fund The interest rate earned by the fund varied between 21% and

so Caperton would like to know the for his earned retirement dollars From a time line (see Figure 16), we set up

hard-th f ll i ti i 1 1 i f C t ’ i

YTM 5 i 23%,

2457.67%.

24.5767 5 0.0767 5 7.67%x 5 23.5767. x 5 1 1 i,

x 5 1.0767

x 522500 6 "2500211000222 4110002 1238512

c 5 23851. a 5 1000,b 5 2500,1000x 2 1 2500x 2 3851 5 0

x 5 1 1 i

1 1 i

Determine whether each of the following statements is true or

false, and explain why.

1 For a particular interest rate, compound interest is always

bet-ter than simple inbet-terest.

2 The sequence 1, 2, 4, 6, 8, is a geometric sequence.

3 If a geometric sequence has first term 3 and common ratio 2,

then the sum of the first 5 terms is

4 The value of a sinking fund should decrease over time.

5 For payments made on a mortgage, the (noninterest) portion of

the payment applied on the principal increases over time.

6 On a 30-year conventional home mortgage, at recent interest

rates, it is common to pay more money on the interest on the

loan than the actual loan itself.

7 One can use the amortization payments formula to calculate

the monthly payment of a car loan.

8 The effective rate formula can be used to calculate the present

value of a loan.

S5 5 93

REVIEW EXERCISES

CONCEPT CHECK

9 The following calculation gives the monthly payment on a

$25,000 loan, compounded monthly at a rate of 5% for a period of six years:

10 The following calculation gives the present value of an

annu-ity of $5,000 payments at the end of each year for 10 years.

The fund earns 4.5% compounded annually.

Find the simple interest for each loan.

PRACTICE AND EXPLORATION

䉴

PREFACE xvii

Flexible Syllabus

The flexibility of the text is indicated in the following chart of chapter prerequisites As shown, the course could begin with either Chapter 1 or Chapter 7 Chapter 5 on the mathe- matics of finance and Chapter 6 on logic could be covered at any time, although Chapter 6 makes a nice introduction to ideas covered in Chapter 7.

CHAPTER 1: Linear Functions

CHAPTER 9: Statistics

CHAPTERS 10–18: Calculus

CHAPTER 5: Mathematics of

Finance CHAPTER 6: Logic

CHAPTER 8: Counting Principles;

Further Probability Topics

CHAPTER 7: Sets and Probability

CHAPTER 2: Systems of Linear

Equations and Matrices

CHAPTER 3: Linear Programming:

The Graphical Method

CHAPTER 4: Linear Programming:

The Simplex Method

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PREFACE xix

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Acknowledgments We wish to thank the following professors for their contributions in reviewing portions of

C.T Bruns, University of Colorado, Boulder James K Bryan, Jr., Merced College Nurit Budinsky, University of Massachusetts—Dartmouth James Carolan, Wharton County Junior College

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Kevin Farrell, Lyndon State College Chris Ferbrache, Fresno City College Lauren Fern, University of Montana Pete Gomez, Houston Community College, Northwest Sharda K Gudehithlu, Wilbur Wright College Mary Beth Headlee, State College of Florida Yvette Hester, Texas A & M University David L Jones, University of Kansas Karla Karstens, University of Vermont Monika Keindl, Northern Arizona University Lynette J King, Gadsden State Community College Jason Knapp, University of Virginia

Donna S Krichiver, Johnson County Community College Mark C Lammers, University of North Carolina, Wilmington Lia Liu, University of Illinois at Chicago

Donna M Lynch, Clinton Community College Rebecca E Lynn, Colorado State University Rodolfo Maglio, Northeastern Illinois University Cyrus Malek, Collin College

Phillip Miller, Indiana University Southeast Marna Mozeff, Drexel University

Javad Namazi, Fairleigh Dickinson University Bishnu Naraine, St Cloud State University Dana Nimic, Southeast Community College—Lincoln Lisa Nix, Shelton State Community College

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Charles B Pierre, Clark Atlanta University Stela Pudar-Hozo, Indiana University Northwest Brooke Quinlan, Hillsborough Community College Candace Rainer, Meridian Community College Nancy Ressler, Oakton Community College Arthur J Rosenthal, Salem State College Theresa Rushing, The University of Tennessee at Martin Katherine E Schultz, Pensacola Junior College Barbara Dinneen Sehr, Indiana University, Kokomo Gordon H Shumard, Kennesaw State University

Walter Sizer, Minnesota State University, Moorhead

Mary Alice Smeal, Alabama State University

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Darren Tapp, Hesser College

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Amanda Wheeler, Amarillo College

Douglas Williams, Arizona State University

Roger Zarnowski, Angelo State University

We also thank Elka Block and Frank Purcell of Twin Prime Editorial for doing an excellent

job updating the Student’s Solutions Manual and Instructor’s Resource Guide and Solutions

Manual, an enormous and time-consuming task Further thanks go to our accuracy

checkers Renato Mirollo, Jon Weerts, Tom Wegleitner, Nathan Kidwell, John Samons, and Lauri Semarne We are very thankful for the work of William H Kazez, Theresa Laurent, and Richard McCall, in writing Extended Applications for the book We are grateful to Karla Harby and Mary Ann Ritchey for their editorial assistance We especially appreciate the staff at Pearson, whose contributions have been very important in bringing this project

to a successful conclusion.

Margaret L Lial Raymond N Greenwell Nathan P Ritchey

PREFACE xxi

Dear Student,

Hello! The fact that you’re reading this preface is good news One of the keys to cess in a math class is to read the book Another is to answer all the questions correctly

suc-on your professor’s tests You’ve already started doing the first; doing the secsuc-ond may

be more of a challenge, but by reading this book and working out the exercises, you’ll

be in a much stronger position to ace the tests One last essential key to success is to go

to class and actively participate.

You’ll be happy to discover that we’ve provided the answers to the odd-numbered cises in the back of the book As you begin the exercises, you may be tempted to imme- diately look up the answer in the back of the book, and then figure out how to get that answer It is an easy solution that has a consequence—you won’t learn to do the exer- cises without that extra hint Then, when you take a test, you will be forced to answer the questions without knowing what the answer is Believe us, this is a lot harder! The learning comes from figuring out the exercises Once you have an answer, look in the back and see if your answer agrees with ours If it does, you’re on the right path If it doesn’t, try to figure out what you did wrong Once you’ve discovered your error, con- tinue to work out more exercises to master the concept and skill.

exer-Equations are a mathematician’s way of expressing ideas in concise shorthand The lem in reading mathematics is unpacking the shorthand One useful technique is to read with paper and pencil in hand so you can work out calculations as you go along When you are baffled, and you wonder, “How did they get that result?” try doing the calculation yourself and see what you get You’ll be amazed (or at least mildly satisfied) at how often that answers your question Remember, math is not a spectator sport You don’t learn math by passively reading it or watching your professor You learn mathematics by doing mathematics.

prob-Finally, if there is anything you would like to see changed in the book, feel free to write to

us at matrng@hofstra.edu or npritchey@ysu.edu We’re constantly trying to make this book even better If you’d like to know more about us, we have Web sites that we invite you to visit: http://people.hofstra.edu/rgreenwell and http://people.ysu.edu/~npritchey.

Marge Lial Ray Greenwell Nate Ritchey

Prerequisite Skills Diagnostic Test

Below is a very brief test to help you recognize which, if any, prerequisite skills you may need to remediate in order to be successful in this course After completing the test, check your answers in the back of the book In addition to the answers, we have also pro- vided the solutions to these problems in Appendix A These solutions should help remind you how to solve the problems For problems 5-26, the answers are followed by refer- ences to sections within Chapter R where you can find guidance on how to solve the problem and/or additional instruction Addressing any weak prerequisite skills now will make a positive impact on your success as you progress through this course.

1 What percent of 50 is 10?

2 Simplify

3 Let x be the number of apples and y be the number of oranges Write the following

state-ment as an algebraic equation: “The total number of apples and oranges is 75.”

4 Let s be the number of students and p be the number of professors Write the following

statement as an algebraic equation: “There are at least four times as many students as professors.”

5 Solve for k:

6 Solve for x:

7 Write in interval notation:

8 Using the variable x, write the following interval as an inequality:

9 Solve for y:

10 Solve for

11 Carry out the operations and simplify:

12 Multiply out and simplify

13 Multiply out and simplify

16 Perform the operation and simplify:

17 Perform the operation and simplify:

25 Rationalize the denominator:

26 Simplify "y22 10y 1 25

the back of the book.

R-1

Algebra Reference

R

An expression such as is a term; the number 9 is the coefficient, p is the variable, and

4 is the exponent The expression means while means and so on Terms having the same variable and the same exponent, such as and are like

terms Terms that do not have both the same variable and the same exponent, such as

and are unlike terms.

A polynomial is a term or a finite sum of terms in which all variables have whole

num-ber exponents, and no variables appear in denominators Examples of polynomials include

Order of Operations Algebra is a language, and you must be familiar with its rules

to correctly interpret algebraic statements The following order of operations have been agreed upon through centuries of usage.

• Expressions in parentheses are calculated first, working from the inside out The

numerator and denominator of a fraction are treated as expressions in parentheses.

• Powers are performed next, going from left to right.

• Multiplication and division are performed next, going from left to right.

• Addition and subtraction are performed last, going from left to right.

For example, in the expression suppose x has the value of 2 We

would evaluate this as follows:

Evaluate 3 raised to a power Perform the multiplication.

Evaluate the power.

In the expression suppose x has the value of 2 We would evaluate this as follows:

Evaluate the numerator and the denominator.

Simplify the fraction.

Adding and Subtracting Polynomials The following properties of real bers are useful for performing operations on polynomials.

5 2

221 31 2 2 1 6

16 8

Properties of Real Numbers

For all real numbers a, b, and c:

R.1 Polynomials R-3

Properties of Real Numbers

One use of the distributive property is to add or subtract polynomials Only like terms may be added or subtracted For example,

and

but the polynomial cannot be further simplified To subtract polynomials, we use

show how to add and subtract polynomials.

Adding and Subtracting Polynomials

Add or subtract as indicated.

(a) SOLUTION Combine like terms.

(b) SOLUTION Multiply each polynomial by the coefficient in front of the polynomial, and then combine terms as before.

(c) SOLUTION Distributing the minus sign and combining like terms yields

TRY YOUR TURN 1

Multiplying Polynomials The distributive property is also used to multiply polynomials, along with the fact that For example,

Multiplying Polynomials

Multiply.

(a) SOLUTION Using the distributive property yields

5 48x22 32x.

8x1 6x 2 4 2 5 8x1 6x 2 2 8x1 4 2 8x1 6x 2 4 2

21 24x41 6x32 9x22 12 2 1 3123x31 8x22 11x 1 7 2

5 11x31 x22 3x 1 8

5 18 x31 3 x32 1 124 x21 5 x22 1 16 x 2 9 x 2 1 8 18x32 4x21 6x2 1 13x31 5x22 9x 1 82

(b) SOLUTION Using the distributive property yields

(c) SOLUTION Multiplying the first two polynomials and then multiplying their product

by the third polynomial yields

. TRY YOUR TURN 2

A binomial is a polynomial with exactly two terms, such as or When two binomials are multiplied, the FOIL method (First, Outer, Inner, Last) is used as a memory aid.

first two factors using FOIL.

Now multiply this last result by using the distributive property, as in Example 3(b).

Combine like terms.

Notice in the first part of Example 5, when we multiplied by itself, that the product of the square of a binomial is the square of the first term, plus twice the prod- uct of the two terms, 1 2 2 1 2k 2 1 25m 2 , plus the square of the last term, 1 2k 2 1 25k 22.

m 1 n 2x 1 1

YOUR TURN 2 Perform the

operation (3y ⫹ 2)(4y2⫺ 2y ⫺ 5).

1 r 1 2s 2 3t 2 1 2r 2 2s 1 t 2

1 x 1 y 1 z 2 1 3x 2 2y 2 z 21k 1 22 112k32 3k21 k 1 1212m 1 12 14m22 2m 1 12

1 3p 1 2 2 1 5p21 p 2 4 213p 2 12 19p21 3p 1 12

Multiplication of polynomials relies on the distributive property The reverse process,

where a polynomial is written as a product of other polynomials, is called factoring For

example, one way to factor the number 18 is to write it as the product both 9 and 2 are

factors of 18 Usually, only integers are used as factors of integers The number 18 can also

be written with three integer factors as

The Greatest Common Factor To factor the algebraic expression

first note that both 15m and 45 are divisible by 15; and By the distributive property,

Both 15 and are factors of Since 15 divides into both terms of

(and is the largest number that will do so), 15 is the greatest common factor for

the polynomial The process of writing as is often called

factoring out the greatest common factor.

Factoring

Factor out the greatest common factor.

(a) SOLUTION Both 12p and 18q are divisible by 6 Therefore,

(b) SOLUTION Each of these terms is divisible by x.

TRY YOUR TURN 1

One can always check factorization by finding the product of the factors and comparing

it to the original expression.

When factoring out the greatest common factor in an expression like becareful to remember the 1 in the second term

Factoring Trinomials A polynomial that has no greatest common factor (other than 1) may still be factorable For example, the polynomial can be factored as

To see that this is correct, find the product you should get

A polynomial such as this with three terms is called a trinomial To factor the

trinomial where the coefficient of is 1, we use FOIL backwards.

Factoring a Trinomial

Factor

SOLUTION Since the coefficient of is 1, factor by finding two numbers whose product is

15 and whose sum is 8 Since the constant and the middle term are positive, the numbers must

both be positive Begin by listing all pairs of positive integers having a product of 15 As you do this, also form the sum of each pair of numbers.

The numbers 5 and 3 have a product of 15 and a sum of 8 Thus, factors as

The answer also can be written as

If the coefficient of the squared term is not 1, work as shown below.

Factoring a Trinomial

Factor

SOLUTION The possible factors of are 4x and x or 2x and 2x; the possible factors of

are and y or 5y and Try various combinations of these factors until one works (if, indeed, any work) For example, try the product

R.2 Factoring R-7

This product is not correct, so try another combination.

Since this combination gives the correct polynomial,

TRY YOUR TURN 2

Special Factorizations Four special factorizations occur so often that they are listed here for future reference.

(e) (f ) (g) (h)

In factoring, always look for a common factor first Since has a mon factor of 4,

com-It would be incomplete to factor it as

,

since each factor can be factored still further To factor means to factor

com-pletely, so that each polynomial factor is prime.

36x22 4y25 1 6x 1 2y 2 1 6x 2 2y 236x22 4y25 41 9x22 y22 5 41 3x 1 y 2 1 3x 2 y 2

36x22 4y2

Difference of two squares

p42 1 5 1 p21 1 2 1 p22 1 2 5 1 p21 1 2 1 p 1 1 2 1 p 2 1 2

Difference of two cubes

8k32 27z35 1 2k 232 1 3z 235 1 2k 2 3z 2 1 4k21 6kz 1 9z22

Sum of two cubes

m31 125 5 m31 535 1 m 1 5 2 1m22 5m 1 252

Difference of two cubes

y32 8 5 y32 235 1 y 2 2 2 1 y21 2y 1 4 2 9y22 24yz 1 16z25 13y 2 4z22

x21 12x 1 36 5 1x 1 622

x21 36

Difference of two squares

Factor each polynomial If a polynomial cannot be factored,

write prime Factor out the greatest common factor as necessary.

Rational Expressions

R.3

Many algebraic fractions are rational expressions, which are quotients of polynomials

with nonzero denominators Examples include

Next, we summarize properties for working with rational expressions.

Properties of Rational Expressions

For all mathematical expressions P, Q, R, and S, with and :

Fundamental property Addition

Subtraction Multiplication Division

When writing a rational expression in lowest terms, we may need to use the fact

Reducing Rational Expressions

Write each rational expression in lowest terms, that is, reduce the expression as much as possible.

P

Q 5

PS QS

a32 2169p22 24p 1 16

s22 10st 1 25t2

z21 14zy 1 49y2

9x21 6410x22 160

9m22 25

x22 6424x41 36x3y 2 60x2y2

R.3 Rational Expressions R-9

(b)

The answer cannot be further reduced. TRY YOUR TURN 1

One of the most common errors in algebra involves incorrect use of the fundamental

property of rational expressions Only common factors may be divided or

“can-celed.” It is essential to factor rational expressions before writing them in lowestterms In Example 1(b), for instance, it is not correct to “cancel” (or cancel k, or

divide 12 by because the additions and subtraction must be performed first.Here they cannot be performed, so it is not possible to divide After factoring, how-ever, the fundamental property can be used to write the expression in lowest terms

Combining Rational Expressions

Perform each operation.

(a) SOLUTION Factor where possible, then multiply numerators and denominators and reduce to lowest terms.

(b) SOLUTION Factor where possible

(c) SOLUTION Use the division property of rational expressions.

Invert and multiply.

Factor and reduce to lowest terms.

(d) SOLUTION As shown in the list of properties, to subtract two rational expressions that have the same denominators, subtract the numerators while keeping the same denominator.

11 5k

9p 2 36

51p 2 42 18

5 3 181 y 1 3 2 6 51 y 1 3 2

3y 1 9

18 5y 1 15 5

(e) SOLUTION These three fractions cannot be added until their denominators are the

same A common denominator into which p, 2p, and 3p all divide is 6p Note that 12p

is also a common denominator, but 6p is the least common denominator Use the

fun-damental property to rewrite each rational expression with a denominator of 6p.

(f ) SOLUTION To find the least common denominator, we first factor each denominator.

Then we change each fraction so they all have the same denominator, being careful to multiply only by quotients that equal 1.

Because the numerator cannot be factored further, we leave our answer in this form We could also multiply out the denominator, but factored form is usually more useful.

TRY YOUR TURN 2

5 42 1 27 1 2

6p

5 42 6p 1

27 6p 1

2 6p

7

p 1 2p 9 1

1 3p 5

YOUR TURN 2 Perform each

of the following operations

3a 1 3b4c .

12

51 a 1 b 2

15p3

9p2 4 6p10p2

9k2

25 53k

6y21 11y 1 43y21 7y 1 4

m42 164m22 16

absolute value equations such as The following properties are used to solve linear equations.

Properties of Equality

For all real numbers a, b, and c:

1 If then Addition property of equality

(The same number may be added

to both sides of an equation.)

2 If then Multiplication property of equality

(Both sides of an equation may be multiplied by the same number.)

Solving Linear Equations

Solve the following equations.

(a) SOLUTION The goal is to isolate the variable Using the addition property of equality yields

25m1

4m

m3m22 14m 1 8

3k2k21 3k 2 22

2k2k22 7k 1 3

341k 2 22

831a 2 12 1

2

a 2 1

EXAMPLE 1

(b) SOLUTION Using the multiplication property of equality yields

or

The following example shows how these properties are used to solve linear equations The goal is to isolate the variable The solutions should always be checked by substitution

in the original equation.

Solving a Linear Equation

Solve

SOLUTION

Distributive property Combine like terms.

Add to both sides.

Add to both sides.

Multiply both sides by

Check by substituting in the original equation The left side becomes and the right side becomes Verify that both of these expressions sim-

Quadratic Equations An equation with 2 as the highest exponent of the variable

is a quadratic equation A quadratic equation has the form where a,

is said to be in standard form.

The simplest way to solve a quadratic equation, but one that is not always applicable, is

by factoring This method depends on the zero-factor property.

SOLUTION First write the equation in standard form.

By the zero-factor property, the product can equal 0 if and only if

Solve each of these equations separately to find that the solutions are and Check these solutions by substituting them in the original equation. TRY YOUR TURN 2

23 / 2.

1 / 3 3r 2 1 5 0 or 2r 1 3 5 0.

13r 2 12 12r 1 32 13r 2 12 12r 1 32 5 0.

6r21 7r 2 3

6r21 7r 2 3 5 0 6r21 7r 5 3.

R.4 Equations R-13

Remember, the zero-factor property requires that the product of two (or more)

factors be equal to zero, not some other quantity It would be incorrect to use the

zero-factor property with an equation in the form forexample

If a quadratic equation cannot be solved easily by factoring, use the quadratic formula.

(The derivation of the quadratic formula is given in most algebra books.)

Quadratic Formula

The solutions of the quadratic equation where are given by

.

Quadratic Formula

Solve by the quadratic formula.

SOLUTION The equation is already in standard form (it has 0 alone on one side of the

equal sign), so the values of a, b, and c from the quadratic formula are easily identified The coefficient of the squared term gives the value of a; here, Also, and

(Be careful to use the correct signs.) Substitute these values into the quadratic formula.

The sign represents the two solutions of the equation To find both of the solutions, first use and then use

The two solutions are 5 and

Notice in the quadratic formula that the square root is added to or subtractedfrom the value of before dividing by 2a.

Simplify the solutions by writing as Substituting for gives

Factor Reduce to lowest terms.

The two solutions are and The exact values of the solutions are and The key on a calculator gives decimal approximations of these solutions (to the nearest thousandth):

*

TRY YOUR TURN 3

NOTE Sometimes the quadratic formula will give a result with a negative number under theradical sign, such as A solution of this type is a complex number Since this textdeals only with real numbers, such solutions cannot be used

Equations with Fractions When an equation includes fractions, first eliminate all denominators by multiplying both sides of the equation by a common denominator, a number that can be divided (with no remainder) by each denominator in the equation When

an equation involves fractions with variable denominators, it is necessary to check all

solutions in the original equation to be sure that no solution will lead to a zero denominator.

Solving Rational Equations

Solve each equation.

(a) SOLUTION The denominators are 10, 15, 20, and 5 Each of these numbers can be divided into 60, so 60 is a common denominator Multiply both sides of the equation by

60 and use the distributive property (If a common denominator cannot be found easily, all the denominators in the problem can be multiplied together to produce one.)

Multiply by the common denominator.

5 4 6 "16 2 4 2

x 5 21242 6 "124222 4112 112

2112

*The symbol <means “is approximately equal to.”

YOUR TURN 3 Solve

z21 6 5 8z

EXAMPLE 6