YOU TRY 3 Graph each inequality using the slope-intercept method. a) y 3 2 x 2 b) 3x y 1 3 Solve a System of Linear Inequalities by Graphing A system of linear inequalities contains two or more linear inequalities. The solution set of a system of linear inequalities consists of all the ordered pairs that make all the inequalities in the system true. To solve such a system, we will graph each inequality on the same axes. The region where the solutions intersect, or overlap, is the solution to the system. We use the following steps to solve a system of linear inequalities: Procedure Solving a System of Linear Inequalities by Graphing 1) Graph each inequality separately on the same axes. Shade lightly. 2) The solution set is the intersection (overlap) of the shaded regions. Heavily shade this region to indicate this is the solution set. EXAMPLE 5 Graph the solution set of the system. x 1 2x 3y 3 Solution To graph x 1, graph the boundary line x 1 as a solid line. The x-values are less than 1 to the left of 1, so shade to the left of the line x 1. Graph 2x 3y 3 as shown in the middle graph. The region shaded purple in the third graph is the intersection of the shaded regions and the solution set of the system. x x y 5 x 1 5 5 5 x y 2x 3y 3 2x 3y 3 y x 1 2 T 3 Any point inside the purple area will satisfy both inequalities. For example, the point (2, 4) is in this region (see the graph). Any point outside this region of intersection will not satisfy both inequalities and is not part of the system’s solution set. One such point is (1, 3). Although we show three separate graphs in this example, it is customary to graph everything on the same axes, shading lightly at fi rst, then to heavily shade the region that is the graph of the solution set. In-Class Example 5 Graph the solution set of the system. x 1 2x y 1 Answer: y The final solution is the set of ordered pairs that lie in the overlapping shaded region. x www.mhhe.com/messersmith SECTION 4.5 Linear Inequalities in Two Variables 291
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