1. Copyright © Cengage Learning. All rights reserved. Systems of Linear Equations and Inequalities in Two Variables 7

2. Copyright © Cengage Learning. All rights reserved. Section 7.5 Solving Systems of Linear Inequalities

3. 3 Objectives Determine whether an ordered pair is a solution to a given linear inequality. Graph a linear inequality in one or two variables. Solve an application involving a linear inequality in two variables. Graph the solution set of a system of linear inequalities in one or two variables. 1 1 2 2 3 3 4 4

4. 4 Objectives Solve an application using a system of linear inequalities. 5 5

5. 5 Determine whether an ordered pair is a solution to a given linear inequality 1.

6. 6 Solution to Linear Inequality in 2 Variables A linear inequality in two variables is an inequality that can be written in one of the following forms: Ax + By > C Ax + By < C Ax + By  C Ax + By  C where A, B, and C are real numbers and A and B are not both 0. Some examples of linear inequalities are 2x – y > –3 y < 3 x + 47  6 x  –2 An ordered pair (x, y) is a solution of an inequality in two variables if a true statement results when the values of x and y are substituted into the inequality.

7. 7 Example Determine whether each ordered pair is a solution of y  x – 5. a. (4, 2) b. (0, –6) c. (5, 0) Solution: a. To determine whether (4, 2) is a solution, we substitute 4 for x and 2 for y. y  x – 5 2  4 – 5 2  –1 Since 2  –1 is a true inequality, (4, 2) is a solution.

8. 8 Example – Solution b. To determine whether (0, –6) is a solution, we substitute 0 for x and –6 for y. y  x – 5 –6  0 – 5 –6  –5 Since –6  –5 is a false inequality, (0, –6) is not a solution. cont’d

9. 9 Example – Solution c. To determine whether (5, 0) is a solution, we substitute 5 for x and 0 for y. y  x – 5 0  5 – 5 0  0 Since 0  0 is a true inequality, (5, 0) is a solution. cont’d

10. 10 Graph a linear inequality in one or two variables 2.

11. 11 Graphing Linear Inequality in 2 Variables The graph of y = x – 5 is a line consisting of the points whose coordinates satisfy the equation. The graph of the inequality y  x – 5 is not a line but rather an area bounded by a line, called a half-plane. The half-plane consists of the points whose coordinates satisfy the inequality. The line serves as the boundary of the half-plane.

12. 12 Example Graph the inequality: y  x – 5 Solution: Because equality is included in the original inequality, we begin by graphing the equation y = x – 5 with a solid line, as in Figure 7-11(a). Figure 7-11(a)

13. 13 Example – Solution Because the graph of y  x – 5 also indicates that y can be greater than x – 5, the coordinates of points other than those shown in Figure 7-11(a) satisfy the inequality. To determine which side of the line to shade, we select a test point and substitute its value into the inequality. A convenient test point is the origin. y  x – 5 0  0 – 5 0  –5 Because 0  –5 is true, the coordinates of the origin satisfy the original inequality. Substitute 0 for x and 0 for y. cont’d

14. 14 Example – Solution In fact, the coordinates of every point on the same side of the line as the origin satisfy the inequality. The graph of y  x – 5 is the half-plane that is shaded in Figure 7-11(b). Figure 7-11(b) cont’d

15. 15 Graphing Linear Inequality in 2 Variables Comment The decision to use a dashed line or solid line is determined by the inequality symbol. If the symbol is, the line is dashed. If it is  or , the line is solid.

16. 16 Solve an application involving a linear inequality in two variables 3.

17. 17 Example – Earning Money Chen has two part-time jobs, one paying $5 per hour and the other paying $6 per hour. He must earn at least $120 per week to pay his expenses while attending college. Write an inequality that shows the various ways he can schedule his time to achieve his goal. Solution: If we let x represent the number of hours Chen works on the first job and y the number of hours he works on the second job, we have

18. 18 Example – Solution The graph of the inequality 5x + 6y  120 is shown in Figure 7-14. Since Chen cannot work a negative number of hours, the graph in the figure has no meaning when either x or y is negative. Figure 7-14 cont’d

19. 19 Example – Solution Any point in the shaded region indicates a possible way Chen can schedule his time and earn $120 or more per week. For example, if he works 20 hours on the first job and 10 hours on the second job, he will earn $5(20) + $6(10) = $100 + $60 = $160 cont’d

20. 20 Graph the solution set of a system of linear inequalities in one or two variables 4.

21. 21 Graphing System of Inequalities in 2 Variables We have seen that the graph of a linear inequality in two variables is a half-plane. Therefore, we would expect the graph of a system of two linear inequalities to contain two half-planes. For example, to solve the system x + y  1 x – y  1 we graph each inequality and then superimpose the graphs on one set of coordinate axes.

22. 22 Graphing System of Inequalities in 2 Variables The graph of x + y  1 includes the graph of the equation x + y = 1 and all points above it. Because the boundary line is included, we draw it with a solid line. See Figure 7-15(a). Figure 7-15(a)

23. 23 Graphing System of Inequalities in 2 Variables The graph of x – y  1 includes the graph of the equation x – y = 1 and all points below it. Because the boundary line is included, we draw it with a solid line. See Figure 7-15(b). Figure 7-15(b)

24. 24 Graphing System of Inequalities in 2 Variables In Figure 7-16(a), we show the result when the graphs are superimposed on one coordinate system. The area that is shaded twice represents the set of solutions of the given system. Any point in the doubly shaded region has coordinates that satisfy both of the inequalities. Figure 7-16(a)

25. 25 Graphing System of Inequalities in 2 Variables Figure 7-16(a) is our “working” graph. Just as we solved compound inequalities in one variable, we shade the various regions of the graph and then interpret the results. The solution is only that part of the graph that is doubly shaded, so we create a new graph, Figure 7-16(b), with only the solution. Figure 7-16(b)

26. 26 Graphing System of Inequalities in 2 Variables To see that this is true, we select a test point, such as point A, that lies in the doubly shaded region and show that its coordinates (4, 1) satisfy both inequalities. x + y  1 x – y  1 4 + 1  1 4 – 1  1 5  1 3  1 Since the coordinates of point A satisfy both inequalities, point A is a solution and therefore all points in the doubly shaded region are solutions.

27. 27 Graphing System of Inequalities in 2 Variables In general, to solve systems of linear inequalities, we will take the following steps. Solving Systems of Inequalities 1. Graph each inequality in the system on the same coordinate axes using solid or dashed lines as appropriate. 2. Find the region where the graphs overlap. 3. Select a test point from the region to verify the solution. 4. Graph only the solution set, if there is one.

28. 28 Example 2x + y < 4 –2x + y > 2 Solution: We graph each inequality on one set of coordinate axes, as in Figure 7-17(a). Figure 7-17(a) Graph the solution set:

29. 29 Example – Solution The graph of 2x + y < 4 includes all points below the line 2x + y = 4. Since the boundary is not included, we draw it as a dashed line. The graph of –2x + y > 2 includes all points above the line –2x + y = 2. Since the boundary is not included, we draw it as a dashed line. The area that is shaded twice represents the set of solutions of the given system. cont’d

30. 30 Example – Solution We create a new graph that reflects only the solution, Figure 7-17(b). Select a test point in the doubly shaded region and show that it satisfies both inequalities. Figure 7-17(b) cont’d

31. 31 Solve an application using a system of linear inequalities 5.

32. 32 Example – Landscaping A man budgets from $300 to $600 for trees and bushes to landscape his yard. After shopping around, he finds that good trees cost $150 and mature bushes cost $75. What combinations of trees and shrubs can he afford to buy? 1.What Am I Asked to Find? The man wants to spend at least $300 but not more than $600 for trees and shrubs.

33. 33 Example – Landscaping 1.Form two inequalities We can let x represent the number of trees purchased and y the number of shrubs purchased. We can then form the following system of inequalities. cont’d

34. 34 Example – Landscaping 3.Solve the system We graph the system 150x + 75y  300 150x + 75y  600 as in Figure 7-20. The coordinates of each point shown in the graph give a possible combination of the number of trees (x) and the number of shrubs (y) that can be purchased. Figure 7-20 cont’d

35. 35 Example – Landscaping These possibilities are (0, 4), (0, 5), (0, 6), (0, 7), (0, 8) (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 0), (2, 1), (2, 2), (2, 3), (2, 4) (3, 0), (3, 1), (3, 2), (4, 0) Only these points can be used, because the man cannot buy part of a tree or part of a shrub. cont’d