368 CHAPTER 6 Algebra, Graphs, and Functions Graphing Systems of Linear Inequalities When two or more linear inequalities are considered simultaneously, the inequalities are called a system of linear inequalities. The solution set of a system of linear inequalities is the set of ordered pairs that satisfy all inequalities in the system. The solution set of a system of linear inequalities may consist of infinitely many ordered pairs. To determine the solution set to a system of linear inequalities, graph each inequality on the same axes. The ordered pair solutions common to all the inequalities are the solution set to the system. If we select a point from the shaded half plane, say ( 3, 2), − and substitute x 3 = − and y 2 = into the inequality we get: 2 ( 3) 2, 2 3 ≥ − + or 2 0. ≥ Since 2 0 ≥ is a true statement, we have confirmed that ( 3, 2) − is an element of the solution set. 7 Now try Exercise 19 4 5 3 3 2 2 1 5 4 21 22 23 24 25 22 23 24 25 y x (0, 0) (23, 23) (1, 21) (2, 2) y , x 1 21 Figure 6.33 Example 3 Graphing an Inequality Graph the inequality y x. < Solution The inequality is strictly “less than,” so the boundary line is not part of the solution set. In graphing the equation y x, = draw a dashed line (Fig. 6.33). Since (0, 0) is on the line, it cannot serve as a test point. Let’s pick the point − (1, 1). < − < y x 1 1 True Since 1 1 − < is a true statement, the solution set is represented by the half-plane containing the point − (1, 1). The solution set is indicated in Fig. 6.33. 7 Now try Exercise 27 GRAPHING A SYSTEM OF LINEAR INEQUALITIES 1. Select one of the inequalities. Mentally substitute the equal sign for the inequality sign and plot points as if you were graphing the equation. 2. If the inequality is < or >, draw a dashed line through the points. If the inequality is ≤ or ≥, draw a solid line through the points. 3. Select a test point not on the line and substitute the x- and y-coordinates into the inequality. If the substitution results in a true statement, shade in the area on the same side of the line as the test point. If the substitution results in a false statement, shade in the area on the opposite side of the line as the test point. 4. Repeat steps 1, 2, and 3 for the other inequality. 5. The intersection of the two shaded areas and any solid line common to both inequalities represents the solution set to the system of inequalities. PROCEDURE Example 4 Graphing a System of Inequalities Graph the following system of inequalities and indicate the solution set. < − + − < y x x y 1 5 Solution Graph both inequalities on the same axes. First draw the graph of y x 1. < − + When drawing the graph, remember to use a dashed line, since the inequality is “less than” (see Fig. 6.34a). Shade the half-plane that satisfies the inequality y x 1. < − +
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