6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 369 x 3 4 5 6 1 –3 –2 –1 2 y , 2x 1 1 y –1 –2 –3 –4 2 3 4 5 –5 (a) (1, 0) (0, 1) 1 Figure 6.34 x 3 4 5 6 1 –3 –2 –1 2 x – y , 5 y –1 –2 –3 –4 2 3 4 5 –5 (b) (1, 0) (5, 0) (0, 1) 1 (0, –5) Solution (3, –2) y , 2x 1 1 Learning Catalytics Keyword: Angel-SOM-6.8 (See Preface for additional details.) Now, on the same axes, shade the half-plane that satisfies the inequality x y 5 − < (see Fig. 6.34b). The solution set is represented by the points common to the two shaded half-planes. These are the points in the region on the graph containing both color shadings. In Fig. 6.34(b), we have indicated this region in green. Fig. 6.34(b) shows that the two lines intersect at − (3, 2). This ordered pair can also be determined by any of the algebraic methods discussed in Section 6.7. The ordered pair − (3, 2) is not part of the solution set. 7 Now try Exercise 29 Example 5 Graphing a System of Linear Inequalities Graph the following system of inequalities and indicate the solution set. − ≥ + < x y x y 4 2 8 2 3 6 Solution Graph the inequality x y 4 2 8. − ≥ Remember to use a solid line because the inequality is “greater than or equal to”; see Fig. 6.35(a). x 3 4 5 6 7 1 21 22 23 2 4x 2 2y > 8 y 21 22 23 24 1 2 3 4 x 5 6 7 1 2 3 21 22 23 4x 2 2y > 8 Solution (3, 0) 2x + 3y < 6 y 21 22 23 24 1 2 3 4 (0, –4) (2, 0) (0, 2) (a) (b) (0, 24) (2, 0) 25 25 4 Figure 6.35 On the same set of axes, draw the graph of x y 2 3 6. + < Use a dashed line since the inequality is “less than”; see Fig. 6.35(b). The solution set is represented by the region of the graph that contains both color shadings and the part of the solid blue line that satisfies the inequality x y 2 3 6. + < Note that the point of intersection of the two lines is not a part of the solution set. 7 Now try Exercise 35
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