Graphing Systems of Linear Inequalities in Two Variables

5.5 Graphing Sstems of Linear Inequalities in Two Variables 5.5 OBJECTIVES 1. Graph a sstem of linear inequalities in two variables 2. Solve an application of a sstem of linear inequalities In Section 5.1, we dealt with finding the solution set of a sstem of linear equations. That solution set represented the points of intersection of the graphs of the equations in the sstem. In this section, we etend that idea to include sstems of linear inequalities. In this case, the solution set is all ordered pairs that satisf each inequalit. The graph of the solution set of a sstem of linear inequalities is then the intersection of the graphs of the individual inequalities. Let s look at an eample. Eample 1 Solving a Sstem b Graphing Solve the following sstem of linear inequalities b graphing 4 2 We start b graphing each inequalit separatel. The boundar line is drawn, and using (0, 0) as a test point, we see that we should shade the half plane above the line in both graphs. 4 NOTE Notice that the boundar line is dashed to indicate that points on the line do not represent solutions. 2 NOTE Points on the lines are not included in the solution set. In practice, the graphs of the two inequalities are combined on the same set of aes, as is shown below. The graph of the solution set of the original sstem is the intersection of the graphs drawn above. 4 2 355

356 CHAPTER 5 SYSTEMS OF LINEAR RELATIONS CHECK YOURSELF 1 Solve the following sstem of linear inequalities b graphing. 2 4 3 Most applications of sstems of linear inequalities lead to bounded regions. This requires a sstem of three or more inequalities, as shown in Eample 2. Eample 2 Solving a Sstem b Graphing Solve the following sstem of linear inequalities b graphing. 2 6 5 2 0 On the same set of aes, we graph the boundar line of each of the inequalities. We then choose the appropriate half planes (indicated b the arrow that is perpendicular to the line) in each case, and we locate the intersection of those regions for our graph. NOTE The vertices of the shaded region are given because the have particular significance in later applications of this concept. Can ou see how the coordinates of the vertices were determined? (2, 2) (2, 0) (4, 1) (5, 0) l 1 l 1 : 2 6 l 2 : 5 l 2 CHECK YOURSELF 2 Solve the following sstem of linear inequalities b graphing. 2 8 0 7 0 Let s epand on Eample 4, Section 5.2, to see an application of our work with sstems of linear inequalities. Consider Eample 3.

GRAPHING SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES SECTION 5.5 357 Eample 3 Solving a Business-Based Application A manufacturer produces a standard model and a delue model of a 25-in. television set. The standard model requires 12 h of labor to produce, and the delue model requires 18 h. The labor available is limited to 360 h per week. Also, the plant capacit is limited to producing a total of 25 sets per week. Draw a graph of the region representing the number of sets that can be produced, given these conditions. As suggested earlier, we let represent the number of standard-model sets produced and the number of delue-model sets. Because the labor is limited to 360 h, we have NOTE The total labor is limited to (or less than or equal to) 360 h. 12 18 360 (1) 12 h per standard set 18 h per delue set The total production, here sets, is limited to 25, so we can write 25 (2) NOTE We have 0 and 0 because the number of sets produced cannot be negative. For convenience in graphing, we divide both members of inequalit (1) b 6, to write the equivalent sstem 2 3 60 25 0 0 We now graph the sstem of inequalities as before. The shaded area represents all possibilities in terms of the number of sets that can be produced. 25 30 NOTE The shaded area is called the feasible region. All points in the region meet the given conditions of the problem and represent possible production options. 20 10 10 20 30 2 3 60

358 CHAPTER 5 SYSTEMS OF LINEAR RELATIONS CHECK YOURSELF 3 A manufacturer produces TVs and CD plaers. The TVs require 10 h of labor to produce and the CD plaers require 20 h. The labor hours available are limited to 300 h per week. Eisting orders require that at least 10 TVs and at least 5 CD plaers be produced per week. Draw a graph of the region representing the possible production options. CHECK YOURSELF ANSWERS 1. 2 4 2. 2 8 3 7 0 0 3. Let be the number of TVs and be the number of CD plaers. The sstem is 10 20 300 10 5 30 20 10 10 20 30

Name 5.5 Eercises Section Date In eercises 1 to 18, solve each sstem of linear inequalities b graphing. 1. 2 4 2. 3 6 3. 3 6 1 6 4 ANSWERS 1. 2. 3. 4. 5. 4. 2 8 5. 3 12 6. 2 8 4 2 3 6 3 2 12 6. 7. 8. 9. 10. 11. 7. 3 2 12 8. 2 6 9. 2 8 2 1 1 2 12. 10. 3 6 11. 2 8 12. 6 1 2 6 0 3 3 0 1 359

ANSWERS 13. 14. 15. 16. 13. 3 6 14. 2 2 15. 4 3 12 4 2 6 4 8 0 0 0 0 0 0 17. 18. 19. 16. 2 8 17. 4 4 18. 3 6 3 2 8 2 4 0 2 4 0 In eercises 19 and 20, draw the appropriate graph. 19. Manufacturing. A manufacturer produces both two-slice and four-slice toasters. The two slice toaster takes 6 h of labor to produce and the four-slice toaster 10 h. The labor available is limited to 300 h per week, and the total production capacit is 40 toasters per week. Draw a graph of the feasible region, given these conditions, in which is the number of two-slice toasters and is the number of four-slice toasters. 40 30 20 10 20 40 60 360

ANSWERS 20. Production. A small firm produces both AM and AM/FM car radios. The AM radios take 15 h to produce, and the AM/FM radios take 20 h. The number of production hours is limited to 300 h per week. The plant s capacit is limited to a total of 18 radios per week, and eisting orders require that at least 4 AM radios and at least 3 AM/FM radios be produced per week. Draw a graph of the feasible region, given these conditions, in which is the number of AM radios and the number of AM/FM radios. 20. 21. 22. 23. 24. 21. When ou solve a sstem of linear inequalities, it is often easier to shade the region that is not part of the solution, rather than the region that is. Tr this method, then describe its benefits. 22. Describe a sstem of linear inequalities for which there is no solution. 23. Write the sstem of inequalities 24. Write the sstem of inequalities whose graph is the shaded region. whose graph is the shaded region. 1 3 4 2 3 3 5 2 1 361

Answers 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 40 30 20 10 20 40 60 2 3 3 5 1 362