941 9.6 Systems of Inequalities and Linear Programming Graphing an Inequality in Two Variables Method 1 If the inequality is or can be solved for y, then the following hold true. • The graph of y 6ƒ1x2 consists of all the points that are below the graph of y = ƒ1x2. • The graph of y 7ƒ1x2 consists of all the points that are above the graph of y = ƒ1x2. Method 2 If the inequality is not or cannot be solved for y, then choose a test point not on the boundary. • If the test point satisfies the inequality, then the graph includes all points on the same side of the boundary as the test point. • If the test point does not satisfy the inequality, then the graph includes all points on the other side of the boundary. For either method, use a solid boundary for a nonstrict inequality 1… or Ú2 or a dashed boundary for a strict inequality 16 or 72. Systems of Inequalities The solution set of a system of inequalities is the intersection of the solution sets of all the inequalities in the system. We find this intersection by graphing the solution sets of all inequalities on the same coordinate axes and identifying, by shading, the region common to all graphs. EXAMPLE 3 Graphing Systems of Inequalities Graph the solution set of each system. (a) x 76 - 2y x2 62y (b) x … 3 y … 0 y Ú x + 1 SOLUTION (a) Figures 19(a) and (b) show the graphs of x 76 - 2y and x2 62y. The methods presented earlier in this chapter can be used to show that the boundaries intersect at the points 12, 22 and A -3, 9 2B. The solution set of the system is shown in Figure 19(c). The points on the boundaries of x 76 - 2y and x2 62y do not belong to the graph of the solution set, so the boundaries are dashed. –2 2 4 6 2 0 x y x2 = 2y x2 < 2y (b) –2 2 4 6 2 0 x y x = 6 – 2y x > 6 – 2y x2 < 2y x2 = 2y (2, 2) Solution set of (–3, )9 2 (c) Figure 19 –2 2 4 6 2 0 x y x = 6 – 2y x > 6 – 2y (a) −10 −2 10 10 The region shaded twice is the solution set of the system in Example 3(a).
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