276 Chapter 3 Systems of Linear Equations and Inequalities 3. Suppose there was one additional constraint imposed on Susan’s study time. She knows she needs to spend at least twice as much time studying chemistry as she does studying psychology. Because the time studying chemistry must be at least twice the time studying psychology, we have This inequality may also be written as Figure 3-23 shows the first quadrant with the constraint y 16 14 12 10 6 4 2 2 4 6 8 10 12 14 16 4. At what point in the feasible region is Susan making the most efficient use of her time for both classes? First and foremost, Susan must make use of all 12 hr. This occurs for x y 12. points along the line Susan will also want to study for both classes with approximately twice as much time devoted to chemistry. Therefore, Susan will be deriving the maximum benefit at the point of intersection of the line 1 x y 12 and the line x. 2 Using the substitution method, replace into the equation Clear fractions. Solve for x. x y 12. 2x x 24 3x 24 x 8 y x 8 To solve for y, substitute into the equation y 4 16 14 12 8 6 4 42 2 4 6 8 10 12 14 16 Therefore, Susan should spend 8 hr studying chemistry and 4 hr studying psychology. Skill Practice 7. A local pet rescue group has a total of 30 cages that can be used to hold cats and dogs. Let x represent the number of cages used for cats, and let y represent the number used for dogs. a. Write a set of inequalities to express the fact that the number of cat and dog cages cannot be negative. b. Write an inequality to describe the constraint on the total number of cages for cats and dogs. c. Graph the system of inequalities to find the feasible region describing the available cages. y 1 2 x. 182 2 x 1 2 x 12 y 1 2 x y y 1 2 x. y 1 2 x. x 2y. Answer 7. a. b. c. x 0 and y 0 x y 30 y 105 5 10 15 20 25 30 35 Number of cats Number of dogs x x y 30 35 30 25 20 15 10 5 5 10 y x 10 2 4 Hours (chemistry) Hours (psychology) y x x + y = 12 1 2 (8, 4) 2 x 8 y x 4 2 2 4 Hours (chemistry) Hours (psychology) 1 2 Figure 3-23
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