1.8 Linear Inequalities in Two Variables

Transcription

1 128 CHAPTER 1 Linear Equations and Inequalities 1.8 Linear Inequalities in Two Variables OBJECTIVES 1 Determine Whether an Ordered Pair Is a to a Linear Inequalit 2 Graph Linear Inequalities Solve Problems Involving Linear Inequalities Preparing for Linear Inequalities in Two Variables Before getting started, take this readiness quiz. If ou get a problem wrong, go back to the section cited and review the material. P1. Determine whether = 4 satisfies the inequalit + 1 Ú 7. [Section 1.4, pp ] P2. Solve the inequalit: [Section 1.4, pp ] In Section 1.4, we solved inequalities in one variable. In this section, we discuss linear inequalities in two variables. 1 Determine Whether an Ordered Pair Is a to a Linear Inequalit Linear inequalities in two variables are inequalities in one of the forms A + B 6 C A + B 7 C A + B C A + B Ú C where A, B, and C are real numbers and A and B are not both zero. If we replace the inequalit smbol with an equal sign, we obtain the equation of a line, A + B = C. The line separates the -plane into two regions, called halfplanes. See Figure 56. Figure 56 A B C A linear inequalit in two variables and is satisfied b an ordered pair (a, b) if, when is replaced b a and is replaced b b, a true statement results. EXAMPLE 1 Determining Whether an Ordered Pair Is a to a Linear Inequalit In Two Variables Determine which of the following ordered pairs are solutions to the linear inequalit (a) (2, 4) (b) 1-, 12 (c) (1, ) (a) Let = 2 and = 4 in the inequalit. If a true statement results, then (2, 4) is a solution to the inequalit. Preparing for...answers P1. Satisfies P2. { 6 -} or (- q, -) = 2, = 4: ? ? ? 7 False Because 10 is not less than 7, the statement is false, so (2, 4) is not a solution to the inequalit.

2 Section 1.8 Linear Inequalities in Two Variables 129 (b) Let = - and = 1 in the inequalit. If a true statement results, then 1-, 12 is a solution to the inequalit. = -, = 1: ? ? 7-8 6? 7 Because -8 is less than 7, the statement is true, so 1-, 12 is a solution to the inequalit. (c) Let = 1 and = in the inequalit. = 1, = : ? 7 + 6? 7 6 6? 7 Because 6 is less than 7, the statement is true, so (1, ) is a solution to the inequalit. 1. If we replace the inequalit smbol in A + B 7 C with an equal sign, we obtain the equation of a line, A + B = C. The line separates the -plane into two regions, called. 2. Determine which of the following ordered pairs are solutions to the linear inequalit -2 + Ú. (a) (4, 1) (b) 1-1, 22 (c) (2, ) (d) (0, 1) To review the distinction between strict and nonstrict inequalities, turn back to page Graph Linear Inequalities Now that we know how to determine whether an ordered pair is a solution to a linear inequalit in two variables, we are prepared to graph linear inequalities in two variables. A graph of a linear inequalit in two variables and consists of all points (, ) whose coordinates satisf the inequalit. The graph of an linear inequalit in two variables ma be obtained b graphing the equation corresponding to the inequalit, using dashes if the inequalit is strict ( or ) and a solid line if the inequalit is nonstrict ( or Ú). This graph will separate the -plane into two half-planes. In each half-plane either all points satisf the inequalit or no points satisf the inequalit. So the use of a single test point is all that is required to obtain the graph of a linear inequalit in two variables. EXAMPLE 2 How to Graph a Linear Inequalit in Two Variables Graph the linear inequalit: Step-b-Step Step 1: We replace the inequalit smbol with an equal sign and graph the corresponding line. If the inequalit is strict ( 6 or 7), graph the line as a dashed line. If the inequalit is nonstrict ( or Ú), graph the line as a solid line. We replace 6 with = to obtain + = 7. We graph the line + = 7 1 = using a dashed line because the inequalit is strict. See Figure 57(a) on the net page.

3 10 CHAPTER 1 Linear Equations and Inequalities Step 2: We select an test point that is not on the line and determine whether the test point satisfies the inequalit. When the line does not contain the origin, it is usuall easiest to choose the origin, (0, 0), as the test point Test Point: (0, 0): ? 7 0 6? 7 Because 0 is less than 7, the point (0, 0) satisfies the inequalit. Therefore, we shade the half-plane containing the point (0, 0). See Figure 57(b). The shaded region represents the solution to the linear inequalit. Figure (0, 7) (0, 7) (0, 0) (a) (b) Onl one test point is needed to obtain the graph of the inequalit. Wh? Consider the inequalit presented in Eamples 1 and 2. We notice that A12, 42 does not satisf the inequalit, while B1-, 12 and C11, 2 do satisf the inequalit. Notice that point A is not in the shaded region of Figure 57(b), while points B and C are in the shaded region. So, if a point does not satisf the inequalit, then none of the points in the half-plane containing that point satisf the inequalit. If a point does satisf the inequalit, then all the points in the half-plane containing the point satisf the inequalit. Below we summarize the steps for graphing a linear inequalit in two variables. An alternative to using test points is to solve the inequalit for. If the inequalit is of the form 7 or Ú, shade above the line. If the inequalit is of the form 6 or, shade below the line. STEPS FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES Step 1: Replace the inequalit smbol with an equal sign and graph the resulting equation. If the inequalit is strict ( or ), use dashes to graph the line; if the inequalit is nonstrict ( or Ú) use a solid line. The graph separates the -plane into two half-planes. Step 2: Select a test point P that is not on the line (that is, select a test point in one of the half-planes). (a) If the coordinates of P satisf the inequalit, then shade the half-plane containing P. (b) If the coordinates of P do not satisf the inequalit, then shade the half-plane that does not contain P.. or False: The graph of a linear inequalit is a line. 4. or False: In a graph of a linear inequalit in two variables with a strict inequalit, the line separating the two half-planes should be dashed. In Problems 5 and 6, graph each linear inequalit

4 Section 1.8 Linear Inequalities in Two Variables 11 EXAMPLE Graphing a Linear Inequalit in Two Variables Graph the linear inequalit: Ú 1 2 Do not use (0, 0) as a test point for equations of the form A + B = 0 because the graph of this equation contains the origin. 1 We replace the inequalit smbol with an equal sign to obtain = We graph the line = 1 2. using a solid line because the inequalit is nonstrict. See Figure 58(a). 2 Now, we select an test point that is not on the line and determine whether the test point satisfies the inequalit. Because the line contains the origin, we decide to use (2, 0) as the test point. Test Point: (2, 0): Ú Ú? 1 2 # 2 0 Ú? 1 False Because 0 is not greater than 1, the point (2, 0) does not satisf the inequalit.therefore, we shade the half-plane that does not contain (2, 0). See Figure 58(b).The shaded region represents the solution to the linear inequalit. Notice the inequalit is of the form Ú, so we shade above the line. Figure 58 (2, 0) (a) (b) 7. Graph the linear inequalit: Solve Problems Involving Linear Inequalities Linear inequalities involving two variables can be used to solve problems in areas such as nutrition, manufacturing, or sales. Let s look at an application of linear inequalities from nutrition. EXAMPLE 4 Saturated Fat Intake Rand reall enjos Wend s Junior Cheeseburgers and Biggie French Fries. However, he knows that his intake of saturated fat during lunch should not eceed 16 grams. Each Junior Cheeseburger contains 6 grams of saturated fat and each Biggie Fries contains grams of saturated fat. (SOURCE: wends.com) (a) Write a linear inequalit that describes Rand s options for eating at Wend s. That is, write an inequalit that represents all the combinations of Junior Cheeseburgers and Biggie Fries that Rand can order. (b) Can Rand eat 2 Junior Cheeseburgers and 1 Biggie Fr during lunch and sta within his allotment of saturated fat? (c) Can Rand eat Junior Cheeseburgers and 1 Biggie Fr during lunch and sta within his allotment of saturated fat? (a) We are going to use the first three steps in the problem-solving strateg given in Section 1.2 on page 61 to help us develop the linear inequalit.

5 12 CHAPTER 1 Linear Equations and Inequalities Step 1: Identif We want to determine the number of Junior Cheeseburgers and Biggie Fries Rand can eat while not eceeding 16 grams of saturated fat. Step 2: Name the Unknowns Let represent the number of Junior Cheeseburgers that Rand eats and let represent the number of Biggie Fries Rand eats. Step : Translate If Rand eats one Junior Cheeseburger, then he will consume 6 grams of saturated fat. If he eats two, then he will consume 12 grams of saturated fat. In general, if he eats Junior Cheeseburgers, he will consume 6 grams of saturated fat. Similar logic for the Biggie Fries tells us that if Rand eats Biggie Fries, he will consume grams of saturated fat. The words cannot eceed impl a inequalit. Therefore, a linear inequalit that describes Rand s options for eating at Wend s is The Model (b) Letting = 2 and = 1, we obtain ? 16 15? 16 Because the inequalit is true, Rand can eat 2 Junior Cheeseburgers and 1 Biggie Fr and remain within the allotment of 16 grams of saturated fat. (c) Letting = and = 1, we obtain ? 16 21? 16 False Because the inequalit is false, Rand cannot eat Junior Cheeseburgers and 1 Biggie Fr and remain within the allotment of 16 grams of saturated fat. 8. Aver is on a diet that requires that he consume no more than 800 calories for lunch. He reall enjos Wend s Chicken Breast filet and Frosties. Each Chicken Breast filet contains 40 calories and each Frost contains 0 calories. (a) Write a linear inequalit that describes Aver s options for eating at Wend s. (b) Can Aver eat 1 Chicken Breast filet and 1 Frost and sta within his allotment of calories? (c) Can Aver eat 2 Chicken Breast filets and 1 Frost and sta within his allotment of calories? 1.8 EXERCISES 1 8. are the s that follow each EXAMPLE Building Skills In Problems 9 12, determine whether the given points are solutions to the linear inequalit. See Objective (a) (0, 1) (a) 12, -12 (b) 1-2, 42 (b) 11, -2 (c) 18, -12 (c) 1-5, Ú (a) 1-4, 22 (a) 11, 02 (b) (0, 2) (c) (0, ) (b) (, 0) (c) 11, 22 In Problems 1 2, graph each inequalit. See Objective Ú Ú Ú Ú Ú