939 9.6 Systems of Inequalities and Linear Programming 9.6 Systems of Inequalities and Linear Programming ■ Linear Inequalities in Two Variables ■ Nonlinear Inequalities in Two Variables ■ Systems of Inequalities ■ Linear Programming Linear Inequalities in Two Variables A line divides a plane into three sets of points: the points of the line itself and the points belonging to the two regions determined by the line. Each of these two regions is a half-plane. In Figure 15, line r divides the plane into three different sets of points: line r, halfplane P, and half-plane Q. The points on r belong neither to P nor to Q. Line r is the boundary of each half-plane. x y P Q r 0 Figure 15 Linear Inequality in Two Variables A linear inequality in two variables is an inequality of the form Ax +By "C, where A, B, and C are real numbers, with A and B not both equal to 0. (The symbol … could be replaced with Ú, 6, or 7.) EXAMPLE 1 Graphing a Linear Inequality in Two Variables Graph 3x - 2y … 6. SOLUTION First graph the boundary, 3x - 2y = 6, as shown in Figure 16. Because the points of the line 3x - 2y = 6 satisfy 3x - 2y … 6, this line is part of the solution set. To decide which half-plane (the one above the line 3x - 2y = 6 or the one below the line) is part of the solution set, solve the original inequality for y. 3x - 2y … 6 -2y … -3x + 6 Subtract 3x. y Ú 3 2 x - 3 Divide by -2, and change … to Ú. For a particular value of x, the inequality will be satisfied by all values of y that are greater than or equal to 3 2 x - 3. Thus, the solution set contains the half-plane above the line, as shown in Figure 17. The graph of a linear inequality is a half-plane, perhaps with its boundary. x 2 –3 y 0 3x – 2y = 6 Figure 16 3x – 2y ≤ 6 x 2 –3 y 0 Figure 17 Reverse the inequality symbol when dividing by a negative number. Coordinates for x and y from the solution set (the shaded region) satisfy the original inequality, while coordinates outside the solution set do not. S Now Try Exercise 11.
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