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1 Full Download Applied Calculus for the Managerial Life and Social Sciences A Brief Approach 0th Edition b Tan Solutions at Edited with the trial version of Foit Advanced PDF Editor remove this notice, visit: sciences-a-brief-approach-0th-edition-b-tan

2 Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach, Ninth Edition Soo T. Tan Publisher: Richard Stratton Assistant Editor: Elizabeth Neustaetter Editorial Assistant: Haeree Chang Media Editor: Andrew Coppola Marketing Manager: Ashle Pickering Marketing Coordinator: Shannon Mers Marketing Communications Manager: Mar Anne Paumo Content Project Manager: Cherll Linthicum Art Director: Vernon T. Boes Print Buer: Jud Inoue Rights Acquisitions Specialist: Dean Dauphinais Production Service: Martha Emr Tet Designer: Diane Beasle Photo Researcher: Bill Smith Group Cop Editor: Barbara Willette Illustrator: Jade Mers, Matri Art Services Cover Designer: Irene Morris Cover Images: Stop Lights: Kelvin Murra/ Stone/Gett Images; Charming House: Michael Melford/National Geographic/ Gett Images; Girl with Umbrella: Nick Koudis/Photodisc/Gett Images Compositor: Graphic World, Inc. 0, 009 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered b the copright herein ma be reproduced, transmitted, stored, or used in an form or b an means graphic, electronic, or mechanical, including but not limited to photocoping, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval sstems, ecept as permitted under Section 07 or 08 of the 976 United States Copright Act, without the prior written permission of the publisher. For product information and technolog assistance, contact us at Cengage Learning Customer & Sales Support, For permission to use material from this tet or product, submit all requests online at Further permissions questions can be ed to permissionrequest@cengage.com Librar of Congress Control Number: 0095 ISBN-: ISBN-0: Brooks/Cole 0 Davis Drive Belmont, CA USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Meico, Brazil, and Japan. Locate our local office at Cengage Learning products are represented in Canada b Nelson Education, Ltd. To learn more about Brooks/Cole, visit Purchase an of our products at our local college store or at our preferred online store Printed in the United States of America

3 PRELIMINARIES Yuri Arcurs 00/Shutterstock.com THE FIRST TWO sections of this chapter contain a brief review of algebra. We then introduce the Cartesian coordinate sstem, which allows us to represent points in the plane in terms of ordered pairs of real numbers. This in turn enables us to compute the distance between two points algebraicall. This chapter also covers straight lines. The slope of a straight line plas an important role in the stud of calculus. How much mone is needed to purchase at least 00,000 shares of the Starr Communications Compan? Corbco, a giant conglomerate, wishes to purchase a minimum of 00,000 shares of the compan. In Eample, page, ou will see how Corbco s management determines how much mone the will need for the acquisition.

4 Full Download Applied Calculus for the Managerial Life and Social Sciences A Brief Approach 0th Edition b Tan Solutions at CHAPTER IES PRELIMINAR- Use this test to diagnose an weaknesses that ou might have in the algebra that ou will need for the calculus material that follows. The review section and eamples that will help ou brush up on the skills necessar to work the problem are indicated after each eercise. The answers follow the test. Diagnostic Test. a. Evaluate the epression: (i) (ii) b. Rewrite the epression using positive eponents onl:. Rationalize the numerator: (Eponents and radicals, Eamples and, pages 6 7). Simplif the following epressions: a b. + (Rationalization, Eample 5, page 7) (Operations with algebraic epressions, Eamples 6 and 7, pages 8 9). Factor completel: a. 6a b ca b c9a b b. 6 (Factoring, Eamples 8 0, pages 9 ) 5. Use the quadratic formula to solve the following equation: 9 6. Simplif the following epressions: 7. Perform the indicated operations and simplif: 6 a. # 6 9 b Perform the indicated operations and simplif: 9. Rationalize the denominator: 0. Solve the inequalities: a. 0 b. a 6 9 b > B 5 B z (The quadratic formula, Eample, pages ) t t t t t a. b. 5 t (Rational epressions, Eample, page 6) (Rational epressions, Eamples and, pages 6 8) a. b. # 5 / 9 (Rational epressions, Eamples and 5, pages 8 9) (Rationalizing algebraic fractions, Eample 6, page 9) (Inequalities, Eample 9, page 0) (Absolute value, Eample, page )

5 . PRECALCULUS REVIEW I ANSWERS:. a. (i) 6 (ii) b z. a. 5 0 b a. a b a b c ac b. 5. t 6. a. b. ; t 7. a. b. 8. a. 5 b a. [, ] b. c, d. Precalculus Review I Sections. and. review some basic concepts and techniques of algebra that are essential in the stud of calculus. The material in this review will help ou work through the eamples and eercises in this book. You can read through this material now and do the eercises in areas where ou feel a little rust, or ou can review the material on an as-needed basis as ou stud the tet. The self-diagnostic test that precedes this section will help ou pinpoint the areas where ou might have an weaknesses. The Real Number Line The real number sstem is made up of the set of real numbers together with the usual operations of addition, subtraction, multiplication, and division. We can represent real numbers geometricall b points on a real number, or coordinate, line. This line can be constructed as follows. Arbitraril select a point on a straight line to represent the number 0. This point is called the origin. If the line is horizontal, then a point at a convenient distance to the right of the origin is chosen to represent the number. This determines the scale for the number line. Each positive real number lies at an appropriate distance to the right of the origin, and each negative real number lies at an appropriate distance to the left of the origin (Figure ). Origin FIGURE The real number line 0 A one-to-one correspondence is set up between the set of all real numbers and the set of points on the number line; that is, eactl one point on the line is associated with each real number. Conversel, eactl one real number is associated with each point on the line. The real number that is associated with a point on the real number line is called the coordinate of that point.

6 CHAPTER PRELIMINARIES Intervals Throughout this book, we will often restrict our attention to subsets of the set of real numbers. For eample, if denotes the number of cars rolling off a plant assembl line each da, then must be nonnegative that is, 0. Further, suppose management decides that the dail production must not eceed 00 cars. Then, must satisf the inequalit More generall, we will be interested in the following subsets of real numbers: open intervals, closed intervals, and half-open intervals. The set of all real numbers that lie strictl between two fied numbers a and b is called an open interval (a, b). It consists of all real numbers that satisf the inequalities a b, and it is called open because neither of its endpoints is included in the interval. A closed interval contains both of its endpoints. Thus, the set of all real numbers that satisf the inequalities a b is the closed interval [a, b]. Notice that square brackets are used to indicate that the endpoints are included in this interval. Half-open intervals contain onl one of their endpoints. Thus, the interval [a, b) is the set of all real numbers that satisf a b, whereas the interval (a, b] is described b the inequalities a b. Eamples of these finite intervals are illustrated in Table. TABLE Finite Intervals Interval Graph Eample Open: a, b a b, 0 Closed: a, b a b, 0 Half-open: a, b a b, 0 Half-open: a, b, a b 0 In addition to finite intervals, we will encounter infinite intervals. Eamples of infinite intervals are the half-lines (a, ), [a, ), (, a), and (, a] defined b the set of all real numbers that satisf a, a, a, and a, respectivel. The smbol, called infinit, is not a real number. It is used here onl for notational purposes. The notation (, ) is used for the set of all real numbers, since b definition, the inequalities hold for an real number. Infinite intervals are illustrated in Table. TABLE Infinite Intervals Interval Graph Eample a, a, 0 a, a, 0, a a, 0, a a, 0

7 Eponents and Radicals Recall that if b is an real number and n is a positive integer, then the epression b n (read b to the power n ) is defined as the number n factors The number b is called the base, and the superscript n is called the power of the eponential epression b n. For eample, If b 0, we define and For eample, 0 and p 0, but the epression 0 0 is undefined. Net, recall that if n is a positive integer, then the epression b /n is defined to be the number that, when raised to the nth power, is equal to b. Thus, Such a number, if it eists, is called the nth root of b, also written n b. If n is even, the nth root of a negative number is not defined. For eample, the square root of (n ) is not defined because there is no real number b such that b. Also, given a number b, more than one number might satisf our definition of the nth root. For eample, both and squared equal 9, and each is a square root of 9. So to avoid ambiguit, we define b /n to be the positive nth root of b whenever it eists. Thus, 9 9 /. That s wh our calculator will give the answer when ou use it to evaluate 9. Net, recall that if p>q (where p and q are positive integers and q 0) is a rational number in lowest terms, then the epression b p/q is defined as the number b /q p or, equivalentl, q b p, whenever it eists. For eample, Epressions involving negative rational eponents are taken care of b the definition Thus, 5 # # # # b n b # b # b #... # b b 0 b /n n b a b / /..88 b p/q b p/q 5/ 5/ / 5 5 The rules defining the eponential epression a n, where a 0, for all rational values of n are given in Table. The first three definitions in Table are also valid for negative values of a. The fourth definition holds for all values of a if n is odd but onl for nonnegative values of a if n is even. Thus, 8 / 8 8 / has no real value. PRECALCULUS REVIEW I 5 a ba ba b 8 7 n is odd. n is even. Finall, it can be shown that a n has meaning for all real numbers n. For eample, using a calculator with a ke, we see that.665.

8 6 CHAPTER PRELIMINARIES TABLE Rules for Defining a n Definition of a n (a 0) Eample Definition of a n (a 0) Eample Integer eponent: If n is a positive integer, then a n a a a... a 5 (n factors of a) (5 factors) Zero eponent: If n is equal to zero, then a (0 0 is not defined.) Negative eponent: If n is a positive integer, then a n a n (a 0) Fractional eponent: a. If n is a positive integer, then a /n or n a denotes the nth root of a. b. If m and n are positive integers, then a m>n n a m n a m c. If m and n are positive integers, then a m/n (a 0) a m/n 6 / 6 8 > 8 9 / 9 / 7 The five laws of eponents are listed in Table. TABLE Laws of Eponents Law Eample. a m a n a mn 5 a m. n amn a 0 a. a m n a mn. ab n a n b n 6 5. b 0 a a a n b b an b b n These laws are valid for an real numbers a, b, m, and n whenever the quantities are defined. Remember, 5. The correct equation is 6. The net several eamples illustrate the use of the laws of eponents. VIDEO EXAMPLE Simplif the epressions: 6 5/ a. b. c. d. e. a / 6 / 6 b / /

9 . PRECALCULUS REVIEW I 7 Solution 5 a. Law 6 5/ b. Law 6 / 65// 6 / 6 8 c. Law 6 / 6 / 6 6 d. Law 6 6 e. a / / / Law 5 / / / b We can also use the laws of eponents to simplif epressions involving radicals, as illustrated in the net eample. EXAMPLE Simplif the epressions. (Assume that,, m, and n are positive.) a. 6 8 b. m n # m 5 n c. Solution a / 6 / # / 8/ b. 7 6 c. 7 6 / 7/ 8 8 / 8 / m n # m 5 n 6m 8 n 6m 8 n / 6 / # m n 6m n If a radical appears in the numerator or denominator of an algebraic epression, we often tr to simplif the epression b eliminating the radical from the numerator or denominator. This process, called rationalization, is illustrated in the net two eamples. EXAMPLE Rationalize the denominator of the epression. Solution # EXAMPLE Epress as a radical and rationalize the denominator of the / epression that ou obtain. Solution EXAMPLE 5 Rationalize the numerator of the epression. Solution / # #

10 8 CHAPTER PRELIMINARIES Operations with Algebraic Epressions In calculus, we often work with algebraic epressions such as / / An algebraic epression of the form a m n, where the coefficient a is a real number and m and n are nonnegative integers, is called a monomial, meaning it consists of one term. For eample, 7 is a monomial. A polnomial is a monomial or the sum of two or more monomials. For eample, 5 are all polnomials. The degree of a polnomial is the highest power m n of the variables that appears in the polnomial. Constant terms and terms containing the same variable factor are called like, or similar, terms. Like terms ma be combined b adding or subtracting their numerical coefficients. For eample, 7 0 and 7 The distributive propert of the real number sstem, ab ac ab c is used to justif this procedure. To add or subtract two or more algebraic epressions, first remove the parentheses and then combine like terms. The resulting epression is written in order of nonincreasing degree from left to right. EXAMPLE 6 a Remove parentheses. Combine like terms. b. t 5t t t 6 t 5t t t 6 t 5t t 6 t 5t t 6 t 5t t 6 t t t Remove parentheses and combine like terms within brackets. Remove brackets. Combine like terms within braces. Remove braces. Observe that when the algebraic epression in Eample 6b was simplified, the innermost grouping smbols were removed first; that is, the parentheses ( ) were removed first, the brackets [ ] second, and the braces {} third. When algebraic epressions are multiplied, each term of one algebraic epression is multiplied b each term of the other. The resulting algebraic epression is then simplified. EXAMPLE 7 Perform the indicated operations: a. 0 b. a 00 8 b a b c. e t e t e t e t e t e t

11 . PRECALCULUS REVIEW I 9 Solution a b. a 00 8 b a b c. e t e t e t e t e t e t e t e 0 e t e 0 e t e t e 0 e 0 Recall that e 0. Certain product formulas that are frequentl used in algebraic computations are given in Table 5. TABLE 5 Some Useful Product Formulas Formula Eample a b a ab b 9 a b a ab b 6 6 a ba b a b Factoring Factoring is the process of epressing an algebraic epression as a product of other algebraic epressions. For eample, b appling the distributive propert, we ma write To factor an algebraic epression, first check to see whether an of its terms have common factors. If the do, then factor out the greatest common factor. For eample, the common factor of the algebraic epression a a 6a is a because a a 6a a a a a aa EXAMPLE 8 Factor out the greatest common factor in each epression: a. t t b. / / c. e e d. / a b / Solution a. t t tt b. / / / c. e e e

12 0 CHAPTER PRELIMINARIES d. / a b / / / / / / / / / Here we select / as the greatest common factor because it is the highest power of in each algebraic term. In particular, observe that / / / /// / Sometimes an algebraic epression ma be factored b regrouping and rearranging its terms so that a common term can be factored out. This technique is illustrated in Eample 9. EXAMPLE 9 Factor: a. a a b b b. 6 Solution a. First, factor the common term a from the first two terms and the common term b from the last two terms. Thus, a a b b a b Since ( ) is common to both terms of the polnomial, we ma factor it out. Hence a b a b b. 6 6 Rearrange terms. Factor out common factors. As we have seen, the first step in factoring a polnomial is to find the common factors. The net step is to epress the polnomial as the product of a constant and/or one or more prime polnomials. Certain product formulas that are useful in factoring binomials and trinomials are listed in Table 6. TABLE 6 Product Formulas Used in Factoring Formula Eample Difference of two squares: a 6 a a Perfect-square trinomial: 8 6 Sum of two cubes: z 7 z z z z 9 Difference of two cubes: 8 6

13 . PRECALCULUS REVIEW I The factors of the second-degree polnomial with integral coefficients p q r are (a b)(c d), where ac p, ad bc q, and bd r. Since onl a limited number of choices are possible, we can use a trial-and-error method to factor polnomials having this form. For eample, to factor, we first observe that the onl possible firstdegree terms are Since the coefficient of is Net, we observe that the product of the constant terms is (). This gives us the following possible factors: Looking once again at the polnomial, we see that the coefficient of is. Checking to see which set of factors ields for the coefficient of, we find that Factors Coefficients of inner terms Coefficients of outer terms ) Coefficients of inner terms Coefficients of outer terms and we conclude that the correct factorization is Outer terms Inner terms Outer terms Inner terms With practice, ou will soon find that ou can perform man of these steps mentall and the need to write out each step will be eliminated. EXAMPLE 0 Factor: a. b. 6 c. t 9t 95 Solution a. Using trial and error, we find that the correct factorization is b. Since each term has the common factor, we have 6 8 Using the trial-and-error method of factorization, we find that 8 Thus, we have 6 c. Since each term has the common factor, we have t 9t 95 t 6t 65 Using the trial-and-error method of factorization, we find that t 6t 65 t 65t Therefore, t 9t 95 t 65t

14 CHAPTER PRELIMINARIES Roots of Polnomial Equations A polnomial equation of degree n in the variable is an equation of the form a n n a n n a 0 0 where n is a nonnegative integer and a 0, a,..., a n are real numbers with a n 0. For eample, the equation is a polnomial equation of degree 5 in. The roots of a polnomial equation are precisel the values of that satisf the given equation.* One wa to find the roots of a polnomial equation is to factor the polnomial and then solve the resulting equation. For eample, the polnomial equation ma be rewritten in the form 0 0 or 0 Since the product of two real numbers can be equal to zero if and onl if one (or both) of the factors is equal to zero, we have 0 0 or 0 from which we see that the desired roots are 0,, and. The Quadratic Formula In general, the problem of finding the roots of a polnomial equation is a difficult one. But the roots of a quadratic equation (a polnomial equation of degree ) are easil found either b factoring or b using the following quadratic formula. Quadratic Formula The solutions of the equation a b c 0 (a 0) are given b b b ac a Note If ou use the quadratic formula to solve a quadratic equation, first make sure that the equation is in the standard form a b c 0. EXAMPLE Solve each of the following quadratic equations: a. 5 0 b. 8 Solution a. The equation is in standard form, with a, b 5, and c. Using the quadratic formula, we find b b ac 5 5 a 5 5 or *In this book, we are interested onl in the real roots of an equation.

15 . PRECALCULUS REVIEW I This equation can also be solved b factoring. Thus, 5 0 from which we see that the desired roots are or, as obtained earlier. b. We first rewrite the given equation in the standard form 8 0, from which we see that a, b, and c 8. Using the quadratic formula, we find That is, the solutions are b b ac a.7 and 8 In this case, the quadratic formula proves quite hand!.7. Eercises In Eercises 6, show the interval on a number line.. (, 6). (, 5]. [, ). c 6 5. (0, ) 6. (, 5] 5, d In Eercises 7, evaluate the epression / 8. 8 / 9. a / b. ca /. 8 b d #. a 75 7 b / / /8 6 / / 8 /. In Eercises, determine whether the statement is true or false. Give a reason for our choice /8 ca b d a 9 6 b / 8 A 7 # a / b # / /. 5 / 5 / 5 In Eercises 8, rewrite the epression using positive eponents onl... s / s 7/ / / (s t) 8. In Eercises 9 5, simplif the epression. (Assume that,, r, s, and t are positive.) 7/ /. 5. / /.. / 5. a 6. 7 b 6 / a b z r n 7. a 8. b a b r 5n # 9 a e e b / 9. # ab r 6 # s t

16 CHAPTER PRELIMINARIES In Eercises 55 58, use the fact that /. and /.7 to evaluate the epression without using a calculator. 55. / / / / In Eercises 59 6, use the fact that 0 /.6 and 0 /.5 to evaluate the epression without using a calculator / / / In Eercises 6 68, rationalize the denominator In Eercises 69 7, rationalize the numerator B z B In Eercises 75 98, perform the indicated operations and/or simplif each epression a b b a A 96. / / / / In Eercises 99 06, factor out the greatest common factor from each epression z 5 6 z 0. 7a a b 9a b 0. / / 0. e e 0. e e te 0.t 00e 0.t t t t 5/ / u/ u / In Eercises 07 0, factor each epression completel ac bc ad bd a b e e a a b a b / 6 / a ab 6b In Eercises 8, perform the indicated operations and simplif each epression... kr R r kr

17 . PRECALCULUS REVIEW II 5 In Eercises 9, find the real roots of each equation b factoring t t a a 0 In Eercises 5 0, solve the equation b using the quadratic formula DISTRIBUTION OF INCOMES The distribution of income in a certain cit can be described b the mathematical model , where is the number of families with an income of or more dollars. a. How man families in this cit have an income of $0,000 or more? b. How man families have an income of $60,000 or more? c. How man families have an income of $50,000 or more? In Eercises, determine whether the statement is true or false. If it is true, eplain wh it is true. If it is false, give an eample to show wh it is false.. If b ac 0, then a b c 0 (a 0) has two real roots.. If b ac 0, then a b c 0 (a 0) has no real roots.. a bb a b a for all real numbers a and b.. Precalculus Review II Rational Epressions Quotients of polnomials are called rational epressions. Eamples of rational epressions are 6 Since rational epressions are quotients in which the variables represent real numbers, the properties of real numbers appl to rational epressions as well, and operations with rational fractions are performed in the same manner as operations with arithmetic fractions. For eample, using the properties of the real number sstem, we ma write where a, b, and c are an real numbers and b and c are not zero. Similarl, using the same properties of real numbers, we ma write after canceling the common factors. An eample of incorrect cancellation is because is not a factor of the numerator. Instead, we need to write ac bc a b # c c a b # a b 5ab, An algebraic fraction is simplified, or in lowest terms, when the numerator and denominator have no common factors other than and and the fraction contains no negative eponents.

18 6 CHAPTER PRELIMINARIES EXAMPLE Simplif the following epressions: a. b. Solution a. b Factor out ( ). Carr out indicated multiplication. Combine like terms. Cancel the common factors. Factor out from the numerator. The operations of multiplication and division are performed with algebraic fractions in the same manner as with arithmetic fractions (Table 7). TABLE 7 Rules of Multiplication and Division: Algebraic Fractions Operation Eample If P, Q, R, and S are polnomials, then Multiplication: P # R PR Q S QS Division: Q, S 0 P Q R S P # S PS Q R QR Q, R, S 0 # # When rational epressions are multiplied and divided, the resulting epressions should be simplified if possible. VIDEO EXAMPLE Perform the indicated operations and simplif: 8 # 6

19 . PRECALCULUS REVIEW II 7 Solution 8 # 6 # For rational epressions, the operations of addition and subtraction are performed b finding a common denominator of the fractions and then adding or subtracting the numerators. Table 8 shows the rules for fractions with equal denominators. TABLE 8 Rules of Addition and Subtraction: Fractions with Equal Denominators Operation Eample If P, Q, and R are polnomials, then Cancel the common factors ( )( ). Addition: P R Q P Q R R Subtraction: P R Q P Q R R R 0 R To add or subtract fractions that have different denominators, first find a common denominator, preferabl the least common denominator (LCD). Then carr out the indicated operations following the procedure described in Table 8. To find the LCD of two or more rational epressions:. Find the prime factors of each denominator.. Form the product of the different prime factors that occur in the denominators. Each prime factor in this product should be raised to the highest power of that factor appearing in the denominators. EXAMPLE Simplif: a. b. h 6 Solution a. LCD ( )( ) Carr out the indicated multiplication. Combine like terms. Factor.

20 8 CHAPTER PRELIMINARIES h h h h h h h b. LCD ( h) Other Algebraic Fractions Remove parentheses. Combine like terms. The techniques used to simplif rational epressions ma also be used to simplif algebraic fractions in which the numerator and denominator are not polnomials, as illustrated in Eample. EXAMPLE Simplif: a. b. Solution a. b. EXAMPLE 5 Perform the given operations and simplif: / a. # 6 b. Solution a. / # # # # 6 LCD for numerator is and LCD for denominator is. n n # 6 6 / 6 /

21 . PRECALCULUS REVIEW II 9 b. 6 / 6 / 6 / / / 6 / 8 6 / Write radicals in eponential form. LCD is ( ) /. Rationalizing Algebraic Fractions When the denominator of an algebraic fraction contains sums or differences involving radicals, we ma rationalize the denominator that is, transform the fraction into an equivalent one with a denominator that does not contain radicals. In doing so, we make use of the fact that a ba b a b a b This procedure is illustrated in Eample 6. EXAMPLE 6 Rationalize the denominator:. Solution Upon multipling the numerator and the denominator b, we obtain # In other situations, it ma be necessar to rationalize the numerator of an algebraic epression. In calculus, for eample, one encounters the following problem. VIDEO h EXAMPLE 7 Rationalize the numerator:. h Solution h h h h # h h h h h h h h # h h h h h h h h

22 0 CHAPTER PRELIMINARIES Inequalities The following properties ma be used to solve one or more inequalities involving a variable. Properties of Inequalities If a, b, and c are an real numbers, then Eample Propert If a b and b c, and 8, so 8. then a c. Propert If a b, then 5, so 5 ; a c b c. that is,. Propert If a b and c 0, 5, and since 0, we have then ac bc. (5)() ()(); that is, 0 6. Propert If a b and c 0,, and since 0, we have then ac bc. ()() ()(); that is, 6. Similar properties hold if each inequalit sign,, between a and b and between b and c is replaced b,, or. Note that Propert sas that an inequalit sign is reversed if the inequalit is multiplied b a negative number. A real number is a solution of an inequalit involving a variable if a true statement is obtained when the variable is replaced b that number. The set of all real numbers satisfing the inequalit is called the solution set. We often use interval notation to describe the solution set. VIDEO EXAMPLE 8 Find the set of real numbers that satisf 5 7. Solution Add 5 to each member of the given double inequalit, obtaining Net, multipl each member of the resulting double inequalit b, ielding 6 Thus, the solution is the set of all values of ling in the interval [, 6). EXAMPLE 9 Solve the inequalit 8 0. Solution Observe that 8 ( )( ), so the given inequalit is equivalent to the inequalit ( )( ) 0. Since the product of two real numbers is negative if and onl if the two numbers have opposite signs, we solve the inequalit ( )( ) 0 b studing the signs of the two factors and. Now, 0 when, and 0 when. Similarl, 0 when, and 0 when. These results are summarized graphicall in Figure. FIGURE Sign diagram for Sign of ( + ) ( )

23 . PRECALCULUS REVIEW II From Figure, we see that the two factors and have opposite signs when and onl when lies strictl between and. Therefore, the required solution is the interval (, ). EXAMPLE 0 Solve the inequalit. 0 Solution The quotient ( )>( ) is strictl positive if and onl if both the numerator and the denominator have the same sign. The signs of and are shown in Figure. FIGURE Sign diagram for Sign of ( + ) ( ) From Figure, we see that and have the same sign if or. The quotient ( )>( ) is equal to zero if. Therefore, the required solution is the set of all in the intervals (, ] and (, ). APPLIED EXAMPLE Stock Purchase The management of Corbco, a giant conglomerate, has estimated that thousand dollars is needed to purchase shares of common stock of Starr Communications. Determine how much mone Corbco needs to purchase at least 00,000 shares of Starr s stock. Solution The amount of mone Corbco needs to purchase at least 00,000 shares is found b solving the inequalit Proceeding, we find Square both sides. so Corbco needs at least $,000,000. (Recall that is measured in thousands of dollars.) Absolute Value 00, , , Absolute Value The absolute value of a number a is denoted b a and is defined b a if a 0 a e a if a 0

24 CHAPTER PRELIMINARIES a a a 0 a FIGURE The absolute value of a number Since a is a positive number when a is negative, it follows that the absolute value of a number is alwas nonnegative. For eample, 5 5 and 5 (5) 5. Geometricall, a is the distance between the origin and the point on the number line that represents the number a (Figure ). Absolute Value Properties If a and b are an real numbers, then Propert 5 Propert 6 Propert 7 Propert 8 a a ab a b a ` b ` a b 0 b a b a b Eample 6 6 ` ` ` ` Propert 8 is called the triangle inequalit. EXAMPLE Evaluate each of the following epressions: a. p 5 b. Solution a. Since 5 0, we see that 0p 5 0 p 5. Therefore, 0p 5 0 p 5 8 p b. Since 0, we see that 0 0. Net, observe that 0, so 0 0. Therefore, EXAMPLE Solve the inequalities and Solution First, we consider the inequalit If 0, then 0 0, so implies 5 in this case. On the other hand, if 0, then 0 0, so implies 5 or 5. Thus, means 5 5 (Figure 5a). To obtain an alternative solution, observe that 0 0 is the distance from the point to zero, so the inequalit implies immediatel that 5 5. FIGURE (a) (b) Net, the inequalit states that the distance from to zero is greater than or equal to 5. This observation ields the result 5 or 5 (Figure 5b). EXAMPLE Solve the inequalit 0 0. Solution The inequalit 0 0 is equivalent to the inequalities (see Eample ). Thus, and. The solution is therefore given b the set of all in the interval [, ] (Figure 6). FIGURE

25 . PRECALCULUS REVIEW II In Eercises 6, simplif the epression. a ab 9b.. ab b t t.. t In Eercises 7 8, perform the indicated operations and simplif each epression t Eercises a b b a # 6 7 # a b a ab b c t tt t / / d a a b 5b e e / / a b c d / / 00 c t 0t 00t 0 t 0t 00t 0 t 0t 00 / / / In Eercises 9, rationalize the denominator of each epression a b a b.. a b a b In Eercises 5 0, rationalize the numerator of each epression In Eercises, determine whether the statement is true or false In Eercises 5 6, find the values of that satisf the inequalit (inequalities) or 5. or 5. and 5. and a a 5 6 d

26 CHAPTER PRELIMINARIES In Eercises 6 7, evaluate the epression p 70. p In Eercises 7 78, suppose a and b are real numbers other than zero and that a b. State whether the inequalit is true or false for all real numbers a and b. 7. b a 0 7. a b 75. a b 76. a b 77. a b 78. a b In Eercises 79 8, determine whether the statement is true or false for all real numbers a and b. 79. a a 80. b b 8. a a 8. a a 8. a b a b 8. a b a b 85. DRIVING RANGE OF A CAR An advertisement for a certain car states that the EPA fuel econom is 0 mpg cit and 7 mpg highwa and that the car s fuel-tank capacit is 8. gal. Assuming ideal driving conditions, determine the driving range for the car from the foregoing data. 86. Find the minimum cost C (in dollars), given that 5C C 87. Find the maimum profit P (in dollars) given that 6P 500 P CELSIUS AND FAHRENHEIT TEMPERATURES The relationship between Celsius ( C) and Fahrenheit ( F) temperatures is given b the formula C 5 F `.6. ` a. If the average temperature range for Montreal during the month of Januar is 5 C 5, find the range in degrees Fahrenheit in Montreal for the same period. b. If the average temperature range for New York Cit during the month of June is 6 F 80, find the range in degrees Celsius in New York Cit for the same period. 89. MEETING SALES TARGETS A salesman s monthl commission is 5% on all sales over $,000. If his goal is to make a commission of at least $6000/month, what minimum monthl sales figures must he attain? 90. MARKUP ON A CAR The markup on a used car was at least 0% of its current wholesale price. If the car was sold for $,00, what was the maimum wholesale price? 9. QUALITY CONTROL PAR Manufacturing manufactures steel rods. Suppose the rods ordered b a customer are manufactured to a specification of 0.5 in. and are acceptable onl if the are within the tolerance limits of 0.9 in. and 0.5 in. Letting denote the diameter of a rod, write an inequalit using absolute value signs to epress a criterion involving that must be satisfied in order for a rod to be acceptable. 9. QUALITY CONTROL The diameter (in inches) of a batch of ball bearings manufactured b PAR Manufacturing satisfies the inequalit What is the smallest diameter a ball bearing in the batch can have? The largest diameter? 9. MEETING PROFIT GOALS A manufacturer of a certain commodit has estimated that her profit in thousands of dollars is given b the epression where (in thousands) is the number of units produced. What production range will enable the manufacturer to realize a profit of at least $,000 on the commodit? 9. CONCENTRATION OF A DRUG IN THE BLOODSTREAM The concentration (in milligrams/cubic centimeter) of a certain drug in a patient s bloodstream t hr after injection is given b 0.t t Find the interval of time when the concentration of the drug is greater than or equal to 0.08 mg/cc. 95. COST OF REMOVING TOXIC POLLUTANTS A cit s main well was recentl found to be contaminated with trichloroethlene (a cancer-causing chemical) as a result of an abandoned chemical dump that leached chemicals into the water. A proposal submitted to the cit council indicated that the cost, in millions of dollars, of removing % of the toic pollutants is If the cit could raise between $5 million and $0 million for the purpose of removing the toic pollutants, what is the range of pollutants that could be epected to be removed? 96. AVERAGE SPEED OF A VEHICLE The average speed of a vehicle in miles per hour on a stretch of Route between

27 . THE CARTESIAN COORDINATE SYSTEM 5 6 A.M. and 0 A.M. on a tpical weekda is approimated b the epression 0t 0t 50 0 t where t is measured in hours, with t 0 corresponding to 6 A.M. Over what interval of time is the average speed of a vehicle less than or equal to 5 mph? 97. AIR POLLUTION Nitrogen dioide is a brown gas that impairs breathing. The amount of nitrogen dioide present in the atmosphere on a certain Ma da in the cit of Long Beach measured in PSI (pollutant standard inde) at time t, where t is measured in hours and t 0 corresponds to 7 A.M., is approimated b t 0.5t.5 Find the time of the da when the amount of nitrogen dioide is greater than or equal to 8 PSI. Source: Los Angeles Times. In Eercises 98 0, determine whether the statement is true or false. If it is true, eplain wh it is true. If it is false, give an eample to show wh it is false. a 98. b c a b a c 99. If a b, then a c b c. 00. a b b a 0. a b b a 0. a b a b. The Cartesian Coordinate Sstem The Cartesian Coordinate Sstem O -ais O P(, ) FIGURE 8 An ordered pair (, ) Origin -ais FIGURE 7 The Cartesian coordinate sstem In Section., we saw how a one-to-one correspondence between the set of real numbers and the points on a straight line leads to a coordinate sstem on a line (a onedimensional space). A similar representation for points in a plane (a two-dimensional space) is realized through the Cartesian coordinate sstem, which is constructed as follows: Take two perpendicular lines, one of which is normall chosen to be horizontal. These lines intersect at a point O, called the origin (Figure 7). The horizontal line is called the -ais, and the vertical line is called the -ais. A number scale is set up along the -ais, with the positive numbers ling to the right of the origin and the negative numbers ling to the left of it. Similarl, a number scale is set up along the -ais, with the positive numbers ling above the origin and the negative numbers ling below it. The number scales on the two aes need not be the same. Indeed, in man applications, different quantities are represented b and. For eample, ma represent the number of cell phones sold, and ma represent the total revenue resulting from the sales. In such cases, it is often desirable to choose different number scales to represent the different quantities. Note, however, that the zeros of both number scales coincide at the origin of the two-dimensional coordinate sstem. We can now uniquel represent a point in the plane in this coordinate sstem b an ordered pair of numbers that is, a pair (, ), where is the first number and is the second. To see this, let P be an point in the plane (Figure 8). Draw perpendiculars from P to the -ais and to the -ais. Then the number is precisel the number that corresponds to the point on the -ais at which the perpendicular through P crosses the -ais. Similarl, is the number that corresponds to the point on the -ais at which the perpendicular through P crosses the -ais. Conversel, given an ordered pair (, ) with as the first number and the second, a point P in the plane is uniquel determined as follows: Locate the point on the -ais represented b the number and draw a line through that point parallel to the -ais. Net, locate the point on the -ais represented b the number and draw a line through that point parallel to the -ais. The point of intersection of these two lines is the point P (see Figure 8).

28 6 CHAPTER PRELIMINARIES In the ordered pair (, ), is called the abscissa, or -coordinate; is called the ordinate, or -coordinate; and and together are referred to as the coordinates of the point P. Letting P(a, b) denote the point with -coordinate a and -coordinate b, we plot the points A(, ), B(, ), C(, ), D(, ), E(, ), F(, 0), and G(0, 5) in Figure 9. The fact that, in general, P(, ) P(, ) is clearl illustrated b points A and E. The aes divide the -plane into four quadrants. Quadrant I consists of the points P(, ) that satisf 0 and 0; Quadrant II, the points P(, ), where 0 and 0; Quadrant III, the points P(, ), where 0 and 0; and Quadrant IV, the points P(, ), where 0 and 0 (Figure 0). B(, ) A(, ) E(, ) Quadrant II (, +) Quadrant I (+, +) F(, 0) 5 O C(, ) D(, ) 6 G(0, 5) FIGURE 9 Several points in the Cartesian plane Quadrant III (, ) Quadrant IV (+, ) FIGURE 0 The four quadrants in the Cartesian plane (, ) d (, ) The Distance Formula One immediate benefit that arises from using the Cartesian coordinate sstem is that the distance between an two points in the plane ma be epressed solel in terms of their coordinates. Suppose, for eample, that, and, are an two points in the plane (Figure ). Then the distance between these two points can be computed b using the following formula. FIGURE The distance d between the points (, ) and (, ) Distance Formula The distance d between two points P, and P, in the plane is given b d () For a proof of this result, see Eercise 8, page. In what follows, we give several applications of the distance formula. EXAMPLE Find the distance between the points (, ) and (, 6). Solution Let P (, ) and P (, 6) be points in the plane. Then we have Using Formula (), we have 6 d

29 . THE CARTESIAN COORDINATE SYSTEM 7 Eplore & Discuss Refer to Eample. Suppose we label the point (, 6) as P and the point (, ) as P. () Show that the distance d between the two points is the same as that obtained earlier. () Prove that, in general, the distance d in Formula () is independent of the wa we label the two points. VIDEO EXAMPLE Let P(, ) denote a point ling on the circle with radius r and center C(h, k) (Figure ). Find a relationship between and. FIGURE A circle with radius r and center C(h, k) r C(h, k) P(, ) Solution B the definition of a circle, the distance between C(h, k) and P(, ) is r. Using Formula (), we have h k r which, upon squaring both sides, gives the equation h k r that must be satisfied b the variables and. A summar of the result obtained in Eample follows. Equation of a Circle An equation of the circle with center C(h, k) and radius r is given b h k r () EXAMPLE Find an equation of the circle with a. Radius and center (, ). b. Radius and center located at the origin. Solution a. We use Formula () with r, h, and k, obtaining or (Figure a). b. Using Formula () with r and h k 0, we obtain or 9 (Figure b). (, ) FIGURE (a) The circle with radius and (b) The circle with radius and center (, ) center (0, 0)

30 8 CHAPTER PRELIMINARIES Eplore & Discuss. Use the distance formula to help ou describe the set of points in the -plane satisfing each of the following inequalities. a. h k r b. h k r c. h k r d. h k r. Consider the equation. a. Show that. b. Describe the set of points (, ) in the -plane satisfing the following equations: (i) (ii) VIDEO APPLIED EXAMPLE Cost of Laing Cable In Figure, S represents the position of a power rela station located on a straight coastal highwa, and M shows the location of a marine biolog eperimental station on an island. A cable is to be laid connecting the rela station with the eperimental station. If the cost of running the cable on land is $.00 per running foot and the cost of running the cable under water is $5.00 per running foot, find the total cost for laing the cable. (feet) M(0, 000) O Q(000, 0) S(0,000, 0) (feet) Solution The length of cable required on land is given b the distance from S to Q. This distance is (0, ), or 8000 feet. Net, we see that the length of cable required underwater is given b the distance from M to Q. This distance is or approimatel 606 feet. Therefore, the total cost for laing the cable is approimatel dollars. FIGURE Cable connecting rela station S to eperimental station M ,000, ,00

31 . THE CARTESIAN COORDINATE SYSTEM 9 Eplore & Discuss In the Cartesian coordinate sstem, the two aes are perpendicular to each other. Consider a coordinate sstem in which the - and -aes are not collinear and are not perpendicular to each other (see the accompaning figure). O. Describe how a point is represented in this coordinate sstem b an ordered pair, of real numbers. Conversel, show how an ordered pair, of real numbers uniquel determines a point in the plane.. Suppose ou want to find a formula for the distance between two points P, and P, in the plane. What is the advantage that the Cartesian coordinate sstem has over the coordinate sstem under consideration? Comment on our answer.. Self-Check Eercises. a. Plot the points A(, ), B(, ), and C(, ). b. Find the distance between the points A and B; between B and C; between A and C. c. Use the Pthagorean Theorem to show that the triangle with vertices A, B, and C is a right triangle.. The figure opposite shows the location of cities A, B, and C. Suppose a pilot wishes to fl from Cit A to Cit C but must make a mandator stopover in Cit B. If the singleengine light plane has a range of 650 miles, can she make the trip without refueling in Cit B? (miles) 00 C(600, 0) B(00, 50) (miles) A(0, 0) Solutions to Self-Check Eercises. can be found on page.. Concept Questions. What can ou sa about the signs of a and b if the point P(a, b) lies in (a) the second quadrant? (b) The third quadrant? (c) The fourth quadrant?. a. What is the distance between P, and P,? b. When ou use the distance formula, does it matter which point is labeled P and which point is labeled P? Eplain.

32 0 CHAPTER PRELIMINARIES. Eercises In Eercises 6, refer to the following figure and determine the coordinates of each point and the quadrant in which it is located. B. A. B. C. D 5. E 6. F In Eercises 7, refer to the following figure. 7. Which point has coordinates (, )? 8. What are the coordinates of point B? 9. Which points have negative -coordinates? 0. Which point has a negative -coordinate and a negative -coordinate?. Which point has an -coordinate that is equal to zero?. Which point has a -coordinate that is equal to zero? In Eercises 0, sketch a set of coordinate aes and plot each point.. (, 5). (, ) 5. (, ) 6. (, ) 7. 8, 7 5 E D B 5 7 C D 6 E C F A A 6 G F 8. a 5, b 9. (.5,.5) 0. (.,.) In Eercises, find the distance between the given points.. (, ) and (, 7). (, 0) and (, ). (, ) and (, 9). (, ) and (0, 6) 5. Find the coordinates of the points that are 0 units awa from the origin and have a -coordinate equal to Find the coordinates of the points that are 5 units awa from the origin and have an -coordinate equal to. 7. Show that the points (, ), (, 7), (6, ), and (0, ) form the vertices of a square. 8. Show that the triangle with vertices (5, ), (, 5), and (5, ) is a right triangle. In Eercises 9, find an equation of the circle that satisfies the given conditions. 9. Radius 5 and center (, ) 0. Radius and center (, ). Radius 5 and center at the origin. Center at the origin and passes through (, ). Center (, ) and passes through (5, ). Center (a, a) and radius a 5. DISTANCE TRAVELED A grand tour of four cities begins at Cit A and makes successive stops at cities B, C, and D before returning to Cit A. If the cities are located as shown in the following figure, find the total distance covered on the tour. C( 800, 800) D( 800, 0) 500 A(0, 0) (miles) 500 B(00, 00) 500 (miles)

33 . THE CARTESIAN COORDINATE SYSTEM 6. DELIVERY CHARGES A furniture store offers free setup and deliver services to all points within a 5-mi radius of its warehouse distribution center. If ou live 0 mi east and mi south of the warehouse, will ou incur a deliver charge? Justif our answer. 7. OPTIMIZING TRAVEL TIME Towns A, B, C, and D are located as shown in the following figure. Two highwas link town A to town D. Route runs from Town A to Town D via Town B, and Route runs from Town A to Town D via Town C. If a salesman wishes to drive from Town A to Town D and traffic conditions are such that he could epect to average the same speed on either route, which highwa should he take to arrive in the shortest time? marine biolog eperimental station on an island. A cable is to be laid connecting the rela station with the eperimental station. If the cost of running the cable on land is $.00/running foot and the cost of running cable under water is $5.00/running foot, find an epression in terms of that gives the total cost for laing the cable. What is the total cost when 500? When 000? (feet) M(0, 000) (miles) C(800, 500) D(00, 500) O Q(, 0) S(0,000, 0) (feet) 000 A (0, 0) B(00, 00) 000 (miles) 8. MINIMIZING SHIPPING COSTS Refer to the figure for Eercise 7. Suppose a fleet of 00 automobiles are to be shipped from an assembl plant in Town A to Town D. The ma be shipped either b freight train along Route at a cost of 66 /mile per automobile or b truck along Route at a cost of 6 /mile per automobile. Which means of transportation minimizes the shipping cost? What is the net savings? 9. CONSUMER DECISIONS Ivan wishes to determine which HDTV antenna he should purchase for his home. The TV store has supplied him with the following information:. Two ships leave port at the same time. Ship A sails north at a speed of 0 mph while Ship B sails east at a speed of 0 mph. a. Find an epression in terms of the time t (in hours) giving the distance between the two ships. b. Using the epression obtained in part (a), find the distance between the two ships hr after leaving port.. Ship A leaves port sailing north at a speed of 5 mph. A half hour later, Ship B leaves the same port sailing east at a speed of 0 mph. Let t (in hours) denote the time Ship B has been at sea. a. Find an epression in terms of t giving the distance between the two ships. b. Use the epression obtained in part (a) to find the distance between the two ships hr after ship A has left port.. WATCHING A ROCKET LAUNCH At a distance of 000 ft from the launch site, a spectator is observing a rocket being launched. Suppose the rocket lifts off verticall and reaches an altitude of ft (see the accompaning figure). Range in Miles VHF UHF Model Price 0 0 A $ B $ C $ D $80 Rocket Ivan wishes to receive Channel 7 (VHF), which is located 5 mi east and 5 mi north of his home, and Channel 8 (UHF), which is located 0 mi south and mi west of his home. Which model will allow him to receive both channels at the least cost? (Assume that the terrain between Ivan s home and both broadcasting stations is flat.) 0. COST OF LAYING CABLE In the following diagram, S represents the position of a power rela station located on a straight coastal highwa, and M shows the location of a Spectator Launching pad 000 ft a. Find an epression giving the distance between the spectator and the rocket. b. What is the distance between the spectator and the rocket when the rocket reaches an altitude of 0,000 ft?