374 CHAPTER 6 Algebra, Graphs, and Functions In Exercises 13–28, graph the inequality. 13. y x 1 > + * 14. y x 2 < − − * 15. y x2 6 ≥ − * 16. y x2 2 < − + * 17. x y 2 3 6 − > * 18. x y2 4 + > * 19. y x 3 4 2 ≥ − * 20. y x 2 3 1 ≤ − + * 21. x y 3 2 6 + < * 22. x y2 2 − + < * 23. x y 5 2 10 + ≥ * 24. x y 3 2 12 − < * 25. y x2 1 ≥ − + * 26. y x3 4 ≤ − * 27. x y 0 + > * 28. x y2 0 + ≤ * In Exercises 29–38, graph the system of linear inequalities and indicate the solution set. 29. > − + < y x x y 4 5 * 30. − ≤ < − x y y x 3 6 4 * 31. ≤ − + − < y x x y 5 3 * 32. + > ≤ + x y y x 4 2 8 1 * 33. − ≤ + ≥ x y x y 3 3 2 4 * 34. + < + ≥ x y x y 4 3 2 6 * 35. + ≥ − ≥ − x y x y 2 4 3 6 * 36. − ≥ − ≤ x y x y 3 2 6 3 * 37. ≥ ≤ x y 1 1 * 38. ≤ ≤ x y 0 0 * In Exercises 39– 42, a feasible region and its vertices are shown. Determine the maximum and minimum values of the given objective function. 39. K x y 6 4 = + y x 5 4 3 2 1 1 2 3 4 5 6 (0, 4) (2, 3) (5, 0) (0, 0) Maximum is 30 at (5, 0), minimum is 0 at (0, 0). 40. K x y 10 8 = + y x 25 20 15 10 5 5 10152025303540 (0, 20) (15, 11) (8, 16) (20, 0) (0, 0) Maximum is 238 at (15, 11), minimum is 0 at (0, 0). In Exercises 43– 48, a set of constraints and a profit function are given. a) Graph the constraints and determine the vertices of the feasible region. b) Use the vertices obtained in part (a) to determine the maximum and minimum profit. 43. + ≤ + ≤ ≥ ≥ = + x y x y x y P x y 5 2 8 0 0 4 3 * 44. + ≤ + ≤ ≥ ≥ = + x y x y x y P x y 2 6 3 2 12 0 0 2 6 * 45. + ≤ + ≤ ≥ ≥ = + x y x y x y P x y 4 3 6 0 0 7 6 * 46. + ≤ + ≤ ≥ ≥ = + x y x y x y P x y 50 3 90 0 0 20 40 * 47. + ≥ + ≤ ≥ ≥ = + x y x y x y P x y 2 3 18 4 2 20 1 4 2.20 1.65 * 48. + ≤ + ≥ ≥ ≤ ≥ = + x y x y x x y P x y 2 14 7 4 28 2 10 1 15.13 9.35 * Problem Solving In Exercises 49 and 50, use a system of linear inequalities. 49. MODELING—Special Diet Ruben is on a special diet. He must consume fewer than 500 calories at a meal that consists of one serving of chicken and one serving of rice. The meal must contain at least 150 calories from each source. a) Using x to represent the number of calories from chicken and y to represent the number of calories from rice, translate the problem into a system of linear inequalities. x y x y 500, 150, 150 + < ≥ ≥ b) Graph the system of linear inequalities and indicate the solution set. Graph calories from chicken on the horizontal axis and calories from rice on the vertical axis. * c) There are about 180 calories in 3 oz of chicken and about 200 calories in 8 oz of rice. Select a point in the solution set. For the point selected, determine the number of ounces of chicken and the number of ounces of rice to be served. One example is (220, 220); approximately 3.7 oz of chicken, 8.8 oz of rice. 50. MODELING—Tablet Sales The bookstore at St. Petersburg College sells Apple tablets and Samsung tablets. Based on demand, it is necessary to stock at least twice as many Apple tablets as Samsung tablets. The costs to the store are $300 for an Apple tablet and $200 for a Samsung tablet. Management wants at least 10 Apple tablets and at least 5 Samsung tablets in inventory at all $ 41. K x y 2 3 = + y x 50 60 40 30 20 10 10 20 30 40 50 60 (10, 40) (20, 10) (10, 20) (50, 30) (50, 10) Maximum is 190 at (50, 30), minimum is 70 at (20, 10). 42. K x y 40 50 = + y x 6 5 4 3 1 2 5 6 7 1 3 4 2 (2, 5) (4, 3) (5, 2) (4, 1) (1, 1) Maximum is 330 at (2, 5), minimum is 90 at (1, 1). *See Instructor Answer Appendix
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