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# Linear Equations and Inequalities : Ch-02 section-2

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### Linear Equations and Inequalities : Ch-02 section-2

1. 1. 2-2 Linear Equations and Inequalities 105 92. Petroleum Consumption. Analyzing data from the United States Energy Department for the period between 1920 and 1960 reveals that petroleum consumption as a percentage of all energy consumed (wood, coal, petro- leum, natural gas, hydro, and nuclear) increased almost linearly. Percentages for this period are given in the table. Year 1920 1930 1940 1950 1960 Consumption (%) 11 22 29 37 44 (A) Use regression analysis to ﬁnd a linear regression function f(x) for this data, where x is the number of years since 1900. (B) Use f(x) to estimate (to the nearest one percent) the percent of petroleum consumption in 1932. In 1956. (C) If we assume that f(x) continues to provide a good description of the percentage of petroleum consumption after 1960, when would this percentage reach 100%? Is this likely to happen? Explain. Section 2-2 Linear Equations and Inequalities Solving Linear Equations Solving Linear Inequalities Solving Equations and Inequalities Involving Absolute Value Application In this section we discuss methods for solving equations and inequalities that involve linear functions. Some problems are best solved using algebraic tech- niques, while others beneﬁt from a graphical approach. Since graphs often give additional insight into relationships, especially in applications, we will usu- ally emphasize graphical techniques over algebraic methods. But you must be cer- tain to master both. There are problems in this section that can only be solved algebraically. Later we will also encounter problems that can only be solved graphically. Solving Linear Equations In the preceding section we found the x intercept of a linear function f(x) ϭ mx ϩ b by solving the equation f(x) ϭ 0. Now we want to apply the same ideas to some more complicated equations. Solving an Equation Algebraically Solve 5x Ϫ 8 ϭ 2x ϩ 1 and check. S o l u t i o n We use the familiar properties of equality to transform the given equation into an equivalent equation that has an obvious solution (see Section A-8). Original equation Add 8 to both sides.5x Ϫ 8 ؉ 8 ϭ 2x ϩ 1 ؉ 8 5x Ϫ 8 ϭ 2x ϩ 1 E X A M P L E 1
2. 2. 106 2 LINEAR AND QUADRATIC FUNCTIONS Combine like terms. Subtract 2x from both sides. Combine like terms. Divide both sides by 3. Simplify. The solution set for this last equation is obvious: Solution set: {3} It follows from the properties of equality that {3} is also the solution set of all the preceding equations in our solution. [Note: If an equation has only one ele- ment in its solution set, we generally use the last equation (in this case, x ϭ 3) rather than set notation to represent the solution.] C h e c k Original equation Substitute x ϭ 3. Simplify each side. A true statement Solve and check: 2x ϩ 1 ϭ 4x ϩ 5 We can also use a graphing utility to solve equations of this type. From a graphical viewpoint, a solution to an equation of the form f(x) ϭ g(x) is an inter- section point of the graphs of f and g. Figure 1 shows a graphical solution to Example 1 using a built-in intersection routine. Most graphing utilities have such a routine (consult your owner’s manual or the graphing utility manual for this text; see the Preface). If yours does not, then use zoom and trace to approximate intersection points. Ϫ10 Ϫ10 10 10 y2 ϭ 2x ϩ 1 y1 ϭ 5x Ϫ 8 FIGURE 1 Graphical solution of 5x Ϫ 8 ϭ 2x ϩ 1. M A T C H E D P R O B L E M 1 7 ⁄ 7 15 Ϫ 8 ՘ 6 ϩ 1 5(3) Ϫ 8 ՘ 2(3) ϩ 1 5x Ϫ 8 ϭ 2x ϩ 1 x ϭ 3 3x 3 ϭ 9 3 3x ϭ 9 5x ؊ 2x ϭ 2x ϩ 9 ؊ 2x 5x ϭ 2x ϩ 9 106 2 LINEAR AND QUADRATIC FUNCTIONS106 2 LINEAR AND QUADRATIC FUNCTIONS
3. 3. 2-2 Linear Equations and Inequalities 107 Explore/Discuss 1 An equation that is true for all permissible values of the variable is called an identity. An equation that is true for some values of the vari- able and false for others is called a conditional equation. Use algebraic and/or graphical techniques to solve each of the following and identify any identities. (A) 2(x Ϫ 4) ϭ 2x Ϫ 8 (B) 2(x Ϫ 4) ϭ 3x Ϫ 12 (C) 2(x Ϫ 4) ϭ 2x Ϫ 12 Solving an Equation with a Variable in the Denominator Solve algebraically and graphically: S o l u t i o n We begin with an algebraic solution. Note that 0 must be excluded from the per- missible values of x because division by 0 is not permitted. To clear the fractions, we multiply both sides of the equation by 3(2x) ϭ 6x, the least common denom- inator (LCD) of all fractions in the equation. (For a discussion of LCDs and how to ﬁnd them, see Section A-4.) x 0 The solution set is { }. Figure 2 shows the graphical solution. Note that to seven dec- imal places. Solve algebraically and graphically: Remark Which solution method should you use—algebraic or graphical? In Example 1, both the algebraic solution and the graphical solution produced the exact solution, 7 3x ϩ 2 ϭ 1 x Ϫ 3 5 M A T C H E D P R O B L E M 2 111 34 Ϸ 3.2647059, 111 34x ϭ 111 34 Ϫ34x ϭ Ϫ111 The equation is now free of fractions. 21 Ϫ 18x ϭ 16x Ϫ 90 6x ؒ 7 2x Ϫ 6x ؒ 3 ϭ 6x ؒ 8 3 Ϫ 6x ؒ 15 x Multiply by 6x, the LCD. This and the next step usually can be done mentally. 6x΂ 7 2x Ϫ 3΃ϭ 6x΂ 8 3 Ϫ 15 x ΃ 7 2x Ϫ 3 ϭ 8 3 Ϫ 15 x 7 2x Ϫ 3 ϭ 8 3 Ϫ 15 x E X A M P L E 2 Ϫ10 Ϫ10 10 10 y2 ϭ 15 x 8 3 Ϫ y1 ϭ 7 2x Ϫ 3 FIGURE 2 Graphical solution of 8 3 Ϫ 15 x . 7 2x Ϫ 3 ϭ
4. 4. 108 2 LINEAR AND QUADRATIC FUNCTIONS x ϭ 3. In Example 2, the algebraic solution again produced the exact solution, x ϭ 111/34, while the graphical solution produced x ϭ 3.2647059, a seven- decimal-place approximation to the solution. Some like to argue that this makes the algebraic method superior to the graphical method. But exact solutions have little relevance to most applications of mathematics and decimal approximations are usually quite satisfactory. We encourage you to choose the method that seems best to you, and when pos- sible, use the other method to conﬁrm your answer. In a simple problem, like Example 1, choose either method. In Example 2, we would recommend the alge- braic method over the graphical method because of the complexity of the graphs. We have not yet studied graphs of functions involving fractions with x in the denominator. It was a fortunate accident that the intersection point was visible in a standard viewing window. We frequently encounter equations involving more than one variable. For example, if L and W are the length and width of a rectangle, respectively, the area of the rectangle is given by (see Fig. 3): A ϭ LW Depending on the situation, we may want to solve this equation for L or W. To solve for W, we simply consider A and L to be constants and W to be a variable. Then the equation A ϭ LW becomes a linear equation in W, which can be solved easily by dividing both sides by L: Solving an Equation with More than One Variable Solve for P in terms of the other variables: A ϭ P ϩ Prt S o l u t i o n Think of A, r, and t as constants. Factor to isolate P. Divide both sides by 1 ϩ rt. Restriction: 1 ϩ rt 0 Solve for r in terms of the other variables: A ϭ P ϩ Prt Solving Linear Inequalities Now we want to turn our attention to inequalities. Any inequality that can be reduced to one of the four forms in (1) is called a linear inequality in one variable. M A T C H E D P R O B L E M 3 P ϭ A 1 ϩ rt A 1 ϩ rt ϭ P A ϭ P(1 ϩ rt) A ϭ P ϩ Prt E X A M P L E 3 W ϭ A L L 0 A ϭ LW W L FIGURE 3 Area of a rectangle.
5. 5. Explore/Discuss 2 2-2 Linear Equations and Inequalities 109 mx ϩ b Ͼ 0 mx ϩ b Ն 0 mx ϩ b Ͻ 0 mx ϩ b Յ 0 As was the case with equations, the solution set of an inequality is the set of all values of the variable that make the inequality a true statement. Each element of the solution set is called a solution. Two inequalities are said to be equivalent if they have the same solution set. Associated with the linear equation and inequalities 3x Ϫ 12 ϭ 0 3x Ϫ 12 Ͻ 0 3x Ϫ 12 Ͼ 0 is the linear function f(x) ϭ 3x Ϫ 12 (A) Graph the function f. (B) From the graph of f determine the values of x for which f(x) ϭ 0 f(x) Ͻ 0 f(x) Ͼ 0 (C) How are the answers to part B related to the solutions of 3x Ϫ 12 ϭ 0 3x Ϫ 12 Ͻ 0 3x Ϫ 12 Ͼ 0 As you discovered in Explore/Discuss 2, solving inequalities graphically is both intuitive and efﬁcient. On the other hand, algebraic methods can become quite com- plicated. So we will emphasize the graphical approach when solving inequalities. Solving a Linear Inequality Solve and graph on a number line: 0.5x ϩ 1 Յ 0 S o l u t i o n The graph of f(x) ϭ 0.5x ϩ 1 is shown in Figure 4. It is clear from the graph that f(x) is negative to the left of x ϭ Ϫ2 and positive to the right. Thus, the solu- tion set of the inequality 0.5x ϩ 1 Յ 0 is x Յ Ϫ2 or (Ϫϱ, Ϫ2] E X A M P L E 4 Linear inequalities (1) Ϫ10 Ϫ10 10 10 FIGURE 4 f(x) ϭ 0.5x ϩ 1.
6. 6. Explore/Discuss 3 110 2 LINEAR AND QUADRATIC FUNCTIONS Figure 5 shows a graph of the solution set on a number line. A similar graph can be produced on most graphing utilities by entering y1 ϭ 0.5x ϩ 1 Յ 0 (Fig. 6). The expression 0.5x ϩ 1 Յ 0 is assigned the value 1 for those values of x that make it a true statement and the value 0 for those values of x that make it a false statement. Solve and graph on a number line: 2x Ϫ 6 Ն 0 Associated with the following equations and inequalities ϩ 4 ϭ x Ϫ 2 ϩ 4 Ͻ x Ϫ 2 ϩ 4 Ͼ x Ϫ 2 are the two linear functions f(x) ϭ ϩ 4 and g(x) ϭ x Ϫ 2 (A) Graph both f and g in the same viewing window. (B) From the graph in part A determine the value(s) of x for which f(x) ϭ g(x) f(x) Ͻ g(x) f(x) Ͼ g(x) (C) How are the answers to part B related to the solutions of ϩ 4 ϭ x Ϫ 2 ϩ 4 Ͻ x Ϫ 2 ϩ 4 Ͼ x Ϫ 2 Most inequalities can be solved graphically. If you need to algebraically manipulate an inequality, Theorem 1 lists the properties that govern operations on inequalities. Ϫ1 2x Ϫ1 2x Ϫ1 2x Ϫ1 2x Ϫ1 2x Ϫ1 2x Ϫ1 2x M A T C H E D P R O B L E M 4 FIGURE 6FIGURE 5 Ϫ10 Ϫ10 10 10 0Ϫ5 5Ϫ2 x]
7. 7. T H E O R E M 1 2-2 Linear Equations and Inequalities 111 INEQUALITY PROPERTIES An equivalent inequality will result and the sense will remain the same, if each side of the original inequality 1. Has the same real number added to or subtracted from it. 2. Is multiplied or divided by the same positive number. An equivalent inequality will result and the sense will reverse, if each side of the original inequality 3. Is multiplied or divided by the same negative number. Note: Multiplication by 0 and division by 0 are not permitted. To gain some experience with these properties, we will solve the next exam- ple two ways, algebraically and graphically. Solving a Double Inequality Solve and graph on a number line: Ϫ3 Յ 4 Ϫ 7x Ͻ 18 S o l u t i o n To solve algebraically, we perform operations on the double inequality until we have isolated x in the middle with a coefﬁcient of 1. Subtract 4 from each member. or or (2) To solve graphically, enter y1 ϭ Ϫ3, y2 ϭ 4 Ϫ 7x, y3 ϭ 18, graph [Fig. 7(a)], and ﬁnd the intersection points [Figs. 7(b) and 7(c)]. It is clear from the graph that y2 is between y1 and y3 for x between Ϫ2 and 1. Since y2 ϭ Ϫ3 at x ϭ 1, we include 1 in the solution set, obtaining the same solution, as shown in equation (2). (a) (b) (c) Ϫ15 Ϫ5 30 5 Ϫ15 Ϫ5 30 5 Ϫ15 Ϫ5 30 5 y3 ϭ 18 y1 ϭ Ϫ3 y2 ϭ 4 Ϫ 7xFIGURE 7 xx 0Ϫ5 1 5Ϫ2 ( ] (Ϫ2, 1]Ϫ2 Ͻ x Յ 11 Ն x Ͼ Ϫ2 Divide each member by Ϫ7 and reverse each inequality. Ϫ7 ؊7 Ն Ϫ7x ؊7 Ͼ 14 ؊7 Ϫ7 Յ Ϫ7x Ͻ 14 Ϫ3 ؊ 4 Յ 4 Ϫ 7x ؊ 4 Ͻ 18 ؊ 4 Ϫ3 Յ 4 Ϫ 7x Ͻ 18 E X A M P L E 5
8. 8. Explore/Discuss 112 2 LINEAR AND QUADRATIC FUNCTIONS Solve algebraically and graphically and graph on a number line: Ϫ3 Ͻ 7 Ϫ 2x Յ 7 Solving Equations and Inequalities Involving Absolute Value Recall the deﬁnition of the absolute value function (see Section 1-4): f(x) ϭ ͉x͉ ϭ (A) Graph the absolute value function f(x) ϭ ͉x͉ and the constant func- tion g(x) ϭ 3 in the same viewing window. (B) From the graph in part A, determine the values of x for which ͉x͉ Ͻ 3 ͉x͉ ϭ 3 ͉x͉ Ͼ 3 (C) Discuss methods for using the deﬁnition of ͉x͉ and algebraic tech- niques to solve part B. The algebraic solution of an equation or inequality involving the absolute value function usually must be broken down into two or more cases. For exam- ple, to solve the equation ͉x Ϫ 4͉ ϭ 2 (3) we consider two cases: x Ϫ 4 ϭ 2 or x Ϫ 4 ϭ Ϫ2 x ϭ 6 x ϭ 2 We can also solve equation (3) graphically by graphing y1 ϭ ͉x Ϫ 4͉ and y2 ϭ 2, and ﬁnding their intersection points, as shown in Figure 8. (a) (b) Algebraic solutions for inequalities involving absolute values can become quite involved. However, as the next example illustrates, even problems that appear to be complicated are easily solved with a graphing utility. Ϫ10 Ϫ10 10 10 Ϫ10 Ϫ10 10 10 y2 ϭ 2 y1 ϭ ͉x Ϫ 4͉FIGURE 8 Graphical solution of ͉x Ϫ 4͉ ϭ 2. Ά Ϫx x if x Ͻ 0 if x Ն 0 M A T C H E D P R O B L E M 5 4
9. 9. 2-2 Linear Equations and Inequalities 113 Solving Absolute Value Problems Graphically Solve graphically. Write solutions in both inequality and interval notation and graph on a number line. (A) ͉2x Ϫ 5͉ Ͼ 4 (B) ͉0.5x ϩ 2͉ Ն 3x Ϫ 5 S o l u t i o n s (A) Graph y1 ϭ ͉2x Ϫ 5͉ and y2 ϭ 4 in the same viewing window and ﬁnd the intersection points (Fig. 9). Examining the graphs in Figure 9, we see if x Ͻ 0.5 or x Ͼ 4.5, then the graph of y1 is above the graph of y2. Thus, the solution is x Ͻ 0.5 or x Ͼ 4.5 Inequality notation (Ϫϱ, 0.5) ʜ (4.5, ϱ)* Interval notation (a) (b) (B) Figure 10 shows the appropriate graphs for the inequality ͉0.5x ϩ 2͉ Ն 3x Ϫ 5. The graph in Figure 10 shows that y1 Ͼ y2 for x Ͻ 2.8. Since y1 ϭ y2 for x ϭ 2.8, we must include this value of x in the solution set: x Յ 2.8 or (Ϫϱ, 2.8] Solve graphically and write solutions in both inequality and interval notation. (A) ͉ ͉ Ն 2 (B) ͉2x Ϫ 5͉ Ͻ 0.4x ϩ 2 Application Break-Even, Profit, and Loss A recording company produces compact discs (CDs). One-time ﬁxed costs for a particular CD are \$24,000, which include costs such as recording, album E X A M P L E 7 2 3x ϩ 1 M A T C H E D P R O B L E M 6 x 0Ϫ5 2.8 5 ] Ϫ10 Ϫ10 10 10 y2 ϭ 3x Ϫ 5 y1 ϭ ͉0.5x ϩ 2͉ Ϫ10 Ϫ10 10 10 Ϫ10 Ϫ10 10 10 y1 ϭ ͉2x Ϫ 5͉ y2 ϭ 4 FIGURE 10FIGURE 9 Ϫ5 0 5 4.50.5 () x E X A M P L E 6 *The symbol ʜ denotes the union operation for sets. See Section A-8 for a discussion of interval notation and set operations.
10. 10. 114 2 LINEAR AND QUADRATIC FUNCTIONS design, and promotion. Variable costs amount to \$5.50 per CD and include the manufacturing, distribution, and royalty costs for each disc actually manufactured and sold to a retailer. The CD is sold to retail outlets for \$8.00 each. (A) Find the level of sales for which the company will break even. Describe verbally and graphically the sales levels that result in a proﬁt and those that result in a loss. (B) Find the sales level that will produce a proﬁt of \$20,000. S o l u t i o n s (A) Let x ϭ Number of CDs sold C ϭ Total cost for producing x CDs R ϭ Revenue (return) on sales of x CDs Now form the cost and revenue functions. C(x) ϭ Fixed costs ϩ Variable costs ϭ 24,000 ϩ 5.5x Cost function R(x) ϭ 8x Revenue function The company will break even when revenue ϭ cost; that is, when R(x) ϭ C(x). The solution to this equation is often referred to as the break-even point. Graphs of both functions and their intersection point are shown in Fig- ure 11. Examining this graph, we see that the company will break even if they sell 9,600 CDs. If they sell more than 9,600 CDs, then revenue is greater than cost, and the company will make a proﬁt. If they sell fewer than 9,600 CDs, then cost is greater than revenue and the company will lose money. These sales levels are illustrated in Figure 12. (B) The proﬁt function for this manufacturer is P(x) ϭ R(x) Ϫ C(x) ϭ 8x Ϫ (24,000 ϩ 5.5x) ϭ 2.5x Ϫ 24,000 FIGURE 12FIGURE 11 x y 20,000 200,000 y ϭ R(x) ϭ 8x y ϭ C(x) ϭ 24,000 ϩ 5.5x Loss Profit Break-even point 9,600 Ϫ50,000 0 200,000 20,000 y2 ϭ 8x y1 ϭ 24,000 ϩ 5.5x
11. 11. Explore/Discuss 5 2-2 Linear Equations and Inequalities 115 The sales level x that will produce a proﬁt of \$20,000 is the solution of the equation P(x) ϭ 20,000. Figure 13 shows a graphical solution of this linear equation. Thus, we see that the company will make a proﬁt of \$20,000 when they sell 17,600 CDs. Repeat Example 7 if ﬁxed costs are \$28,000, variable costs are \$6.60 per CD, and the CDs are sold for \$9.80 each. (A) Find the x intercept of the proﬁt function in Example 7 (see Fig. 13). (B) Discuss the relationship between the x intercept of the proﬁt func- tion and the sales levels for which the company incurs a loss, breaks even, or makes a proﬁt. (C) In general, compare the graphical solutions of the inequalities f(x) Ͼ g(x) and f(x) Ϫ g(x) Ͼ 0 Answers to Matched Problems 1. Ϫ2 2. Ϸ Ϫ0.5128205 3. r ϭ 4. x Ն 3 or [3, ϱ) 5. 0 Յ x Ͻ 5 or [0, 5) 6. (A) x Յ Ϫ4.5 or x Ն 1.5; (Ϫϱ, Ϫ4.5] ʜ [1.5, ϱ) (B) 1.25 Ͻ x Ͻ 4.375; (1.25, 4.375) 7. (A) The company breaks even if they sell 8,750 CDs, makes a proﬁt if they sell more than 8,750 CDs, and loses money if they sell less than 8,750 CDs. (B) The company must sell 15,000 CDs to make a proﬁt of \$20,000. xx 0 5Ϫ5 1.25 4.375 ( ) 0 1.5 5 Ϫ4.5 ] [ xx Ϫ5 xx 0Ϫ5 5 )[x 0Ϫ5 3 5 [ x A Ϫ P Pt Ϫ 20 39 M A T C H E D P R O B L E M 7 Ϫ50,000 0 50,000 20,000 y3 ϭ 2.5x Ϫ 24,000 y4 ϭ 20,000FIGURE 13 x y 20,0008,750 200,000 Loss Profit Break-even point y ϭ C(x) ϭ 28,000 ϩ 6.6x y ϭ R(x) ϭ 9.8x
12. 12. 116 2 LINEAR AND QUADRATIC FUNCTIONS EXERCISE 2-2 A Use the graphs of functions u and v in the ﬁgure to solve the equations and inequalities in Problems 1–8. (Assume the graphs continue as indicated beyond the portions shown here.) Express solutions to inequalities in interval notation. 1. u(x) ϭ 0 2. v(x) ϭ 0 3. u(x) ϭ v(x) 4. u(x) Ϫ v(x) ϭ 0 5. u(x) Ͼ 0 6. v(x) Ն 0 7. v(x) Ն u(x) 8. v(x) Ͻ 0 Solve Problems 9–14 algebraically and check graphically. 9. 3(x ϩ 2) ϭ 5(x Ϫ 6) 10. 5x ϩ 10(x Ϫ 2) ϭ 40 11. 5 ϩ 4(t Ϫ 2) ϭ 2(t ϩ 7) ϩ 1 12. 5w Ϫ (7w Ϫ 4) Ϫ 2 ϭ 5 Ϫ (3w ϩ 2) 13. 5 Ϫ 14. Solve Problems 15–20 algebraically and check graphically. Represent each solution using inequality notation, interval notation, and a graph on a real number line. 15. 7x Ϫ 8 Ͻ 4x ϩ 7 16. 4x ϩ 8 Ն x Ϫ 1 17. Ϫ5t Ͻ Ϫ10 18. Ϫ7n Ն 21 19. Ϫ4 Ͻ 5t ϩ 6 Յ 21 20. 2 Յ 3m Ϫ 7 Ͻ 14 B In Problems 21–36, solve each equation or inequality. When applicable, write answers using both inequality notation and interval notation. 21. ͉y Ϫ 5͉ ϭ 3 22. ͉x ϩ 1͉ ϭ 5 23. ͉5t ϩ 3͉ Յ 7 24. ͉2w Ϫ 9͉ Ͻ 6 25. 26. 27. Ϫ12 Ͻ (2 Ϫ x) Յ 24 28. 24 Յ (x Ϫ 5) Ͻ 36 2 3 3 4 2 3x ϩ 1 2 ϭ 4 x ϩ 4 3 1 m Ϫ 1 9 ϭ 4 9 Ϫ 2 3m x ϩ 3 4 Ϫ x Ϫ 4 2 ϭ 3 8 2x Ϫ 1 4 ϭ x ϩ 2 3 x y y ϭ u(x) y ϭ v(x) e fdc b a 29. ͉3t Ϫ 7͉ ϭ 30. ͉2s ϩ 3͉ ϭ 6 Ϫ 0.5s 31. ͉1.5x ϩ 6͉ Ͼ 0.3x ϩ 7.5 32. ͉7 Ϫ 2x͉ Ն x Ϫ 0.8 33. 34. 35. 6 Ͻ ͉x Ϫ 2͉ ϩ ͉x ϩ 1͉ Ͻ 12 36. ͉x ϩ 1͉ Ϫ ͉x Ϫ 2͉ Ͻ 0.4x In Problems 37–44, solve for the indicated variable in terms of the other variables. 37. an ϭ a1 ϩ (n Ϫ 1)d for d (arithmetic progressions) 38. F ϭ ϩ 32 for C (temperature scale) 39. for f (simple lens formula) 40. for R1 (electric circuit) 41. A ϭ 2ab ϩ 2ac for a (surface area of a rectangular solid) 42. A ϭ 2ab ϩ 2ac ϩ 2bc for c 43. 44. 45. Discuss the relationship between the graphs of y1 ϭ x and y2 ϭ 46. Discuss the relationship between the graphs of y1 ϭ ͉x͉ and y2 ϭ 47. Discuss the possible signs of the numbers a and b given that (A) ab Ͼ 0 (B) ab Ͻ 0 (C) (D) 48. Discuss the possible signs of the numbers a, b, and c given that (A) abc Ͼ 0 (B) (C) (D) In Problems 49–52, replace each question mark with Ͻ or Ͼ and explain why your choice makes the statement true. 49. If a Ϫ b ϭ 1, then a ? b. 50. If u Ϫ v ϭ Ϫ2, then u ? v. 51. If a Ͻ 0, b Ͻ 0, and Ͼ 1, then a ? b. 52. If a Ͼ 0, b Ͼ 0, and Ͼ 1, then a ? b. b a b a a2 bc Ͻ 0 a bc Ͼ 0 ab c Ͻ 0 a b Ͻ 0 a b Ͼ 0 ͙x2 . ͙x2 . x ϭ 3y ϩ 2 y Ϫ 3 for yy ϭ 2x Ϫ 3 3x ϩ 5 for x 1 R ϭ 1 R1 ϩ 1 R2 1 f ϭ 1 d1 ϩ 1 d2 9 5 C 2x x ϩ 4 ϭ 7 Ϫ 6 x ϩ 4 2x x Ϫ 3 ϭ 7 ϩ 4 x Ϫ 3 4 3 t ϩ 1 2 116 2 LINEAR AND QUADRATIC FUNCTIONS
13. 13. 2-2 Linear Equations and Inequalities 117 C Problems 53–56 are calculus-related. Solve and graph. Write each solution using interval notation. 53. 0 Ͻ ͉x Ϫ 3͉ Ͻ 0.1 54. 0 Ͻ ͉x Ϫ 5͉ Ͻ 0.01 55. 0 Ͻ ͉x Ϫ c͉ Ͻ d 56. 0 Ͻ ͉x Ϫ 4͉ Ͻ d 57. What are the possible values of ? 58. What are the possible values of ? APPLICATIONS 59. Sales Commissions. One employee of a computer store is paid a base salary of \$2,150 a month plus an 8% commis- sion on all sales over \$7,000 during the month. How much must the employee sell in 1 month to earn a total of \$3,170 for the month? 60. Sales Commissions. A second employee of the computer store in Problem 59 is paid a base salary of \$1,175 a month plus a 5% commission on all sales during the month. (A) How much must this employee sell in 1 month to earn a total of \$3,170 for the month? (B) Determine the sales level where both employees receive the same monthly income. If employees can select either of these payment methods, how would you advise an employee to make this selection? 61. Approximation. The area A of a region is approximately equal to 12.436. The error in this approximation is less than 0.001. Describe the possible values of this area both with an absolute value inequality and with interval notation. 62. Approximation. The volume V of a solid is approxi- mately equal to 6.94. The error in this approximation is less than 0.02. Describe the possible values of this volume both with an absolute value inequality and with interval notation. 63. Break-Even Analysis. An electronics ﬁrm is planning to market a new graphing calculator. The ﬁxed costs are \$650,000 and the variable costs are \$47 per calculator. The wholesale price of the calculator will be \$63. For the company to make a proﬁt, it is clear that revenues must be greater than costs. (A) How many calculators must be sold for the company to make a proﬁt? (B) How many calculators must be sold for the company to break even? (C) Discuss the relationship between the results in parts A and B. Խx Ϫ 1Խ x Ϫ 1 x ԽxԽ 64. Break-Even Analysis. A video game manufacturer is planning to market a 64-bit version of its game machine. The ﬁxed costs are \$550,000 and the variable costs are \$120 per machine. The wholesale price of the machine will be \$140. (A) How many game machines must be sold for the company to make a proﬁt? (B) How many game machines must be sold for the company to break even? (C) Discuss the relationship between the results in parts A and B. 65. Break-Even Analysis. The electronics ﬁrm in Problem 63 ﬁnds that rising prices for parts increases the variable costs to \$50.5 per calculator. (A) Discuss possible strategies the company might use to deal with this increase in costs. (B) If the company continues to sell the calculators for \$63, how many must they sell now to make a proﬁt? (C) If the company wants to start making a proﬁt at the same production level as before the cost increase, how much should they increase the wholesale price? 66. Break-Even Analysis. The video game manufacturer in Problem 64 ﬁnds that unexpected programming problems increases the ﬁxed costs to \$660,000. (A) Discuss possible strategies the company might use to deal with this increase in costs. (B) If the company continues to sell the game machines for \$140, how many must they sell now to make a proﬁt? (C) If the company wants to start making a proﬁt at the same production level as before the cost increase, how much should they increase the wholesale price? 55 67. Signiﬁcant Digits. If N ϭ 2.37 represents a measurement, then we assume an accuracy of 2.37 Ϯ 0.005. Express the accuracy assumption using an absolute value inequality. 5 68. Signiﬁcant Digits. If N ϭ 3.65 ϫ 10Ϫ3 is a number from a measurement, then we assume an accuracy of 3.65 ϫ 10Ϫ3 Ϯ 5 ϫ 10Ϫ6 . Express the accuracy assumption using an absolute value inequality. 5 69. Finance. If an individual aged 65–69 continues to work after Social Security beneﬁts start, beneﬁts will be reduced when earnings exceed an earnings limitation. In 1989, beneﬁts were reduced by \$1 for every \$2 earned in excess of \$8,880. Find the range of beneﬁt reductions for individ- uals earning between \$13,000 and \$16,000. 5 70. Finance. Refer to Problem 69. In 1990 the law was changed so that beneﬁts were reduced by \$1 for every \$3 earned in excess of \$8,880. Find the range of beneﬁt re- ductions for individuals earning between \$13,000 and \$16,000.
14. 14. 118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius de- grees to Fahrenheit degrees is given by the linear function F ϭ ϩ 32 Determine to the nearest degree the Celsius range in tem- perature that corresponds to the Fahrenheit range of 60°F to 80°F. 72. Celsius/Fahrenheit. A formula for converting Fahrenheit degrees to Celsius degrees is given by the linear function C ϭ (F Ϫ 32) Determine to the nearest degree the Fahrenheit range in temperature that corresponds to a Celsius range of 20°C to 30°C. 5 73. Earth Science. In 1984, the Soviets led the world in drilling the deepest hole in the Earth’s crust—more than 12 kilometers deep. They found that below 3 kilometers the temperature T increased 2.5°C for each additional 100 meters of depth. (A) If the temperature at 3 kilometers is 30°C and x is the depth of the hole in kilometers, write an equation using x that will give the temperature T in the hole at any depth beyond 3 kilometers. (B) What would the temperature be at 15 kilometers? [The temperature limit for their drilling equipment was about 300°C.] 5 9 9 5 C (C) At what interval of depths will the temperature be between 200°C and 300°C, inclusive? 5 74. Aeronautics. Because air is not as dense at high altitudes, planes require a higher ground speed to become airborne. A rule of thumb is 3% more ground speed per 1,000 feet of elevation, assuming no wind and no change in air tempera- ture. (Compute numerical answers to 3 signiﬁcant digits.) (A) Let Vs ϭ Takeoff ground speed at sea level for a particular plane (in miles per hour) A ϭ Altitude above sea level (in thousands of feet) V ϭ Takeoff ground speed at altitude A for the same plane (in miles per hour) Write a formula relating these three quantities. (B) What takeoff ground speed would be required at Lake Tahoe airport (6,400 feet), if takeoff ground speed at San Francisco airport (sea level) is 120 miles per hour? (C) If a landing strip at a Colorado Rockies hunting lodge (8,500 feet) requires a takeoff ground speed of 125 miles per hour, what would be the takeoff ground speed in Los Angeles (sea level)? (D) If the takeoff ground speed at sea level is 135 miles per hour and the takeoff ground speed at a mountain resort is 155 miles per hour, what is the altitude of the mountain resort in thousands of feet? Section 2-3 Quadratic Functions Quadratic Functions Completing the Square Properties of Quadratic Functions and Their Graphs Applications Quadratic Functions The graph of the square function, h(x) ϭ x2 , is shown in Figure 1. Notice that the graph is symmetric with respect to the y axis and that (0, 0) is the lowest point on the graph. Let’s explore the effect of applying a sequence of basic transfor- mations to the graph of h. (A brief review of Section 1-5 might prove helpful at this point.) h(x) 5 Ϫ5 5 x FIGURE 1 Square function h(x) ϭ x2 .