Graphing a Linear Inequality in Two Variables

1. Replace the inequality symbol with an equal sign and graph the corresponding linear equation. Draw a solid line if the original inequality contains a < or > symbol. Draw a dashed line if the original inequality contains a < or > symbol. • Choose a test point in one of the half-planes that is not on the line. Substitute the coordinates of the test point into the inequality. • If a true statement results, shade the half-plane containing this test point. If a false statement results, shade the half-plane not containing this test point. Graphing a Linear Inequality in Two Variables

2. Solution Step 1Replace the inequality symbol by = and graph the linear equation. We need to graph 3x – 5y = 15. We can use intercepts to graph this line. We set y = 0 to find We set x = 0 to find the x-intercept: the y-intercept: 3x – 5y = 15 3x – 5y = 15 3x – 5 • 0 = 15 3 – 0 • 5y = 15 3x = 15 -5y = 15 x = 5 y = -3 Graph: 3x – 5y < 15. Example: Graphing a Linear Inequality in Two Variables

3. 5 4 3 2 1 -3 -2 -1 1 2 3 4 5 6 7 -1 -2 -3 -4 3x – 5y = 15 -5 Solution The x-intercept is 5, so the line passes through (5, 0). The y-intercept is -3, so the line passes through (0, -3). The graph is indicated by a dashed line because the inequality 3x – 5y < 15 contains a < symbol, rather than <. The graph of the line is shown below.

4. Solution Step 2Choose a test point in one of the half-planes that is not on the line. Substitute its coordinates into the inequality. The line 3x – 5y = 15 divides the plane into three parts – the line itself and two half-planes. The points in one half-plane satisfy 3x – 5y > 15. The points in the other half-plane satisfy 3x – 5y < 15. We need to find which half-plane is the solution. To do so, we test a point from either half-plane. The origin, (0, 0), is the easiest point to test. 3x – 5y < 15 This is the given inequality. Is 3 • 0 – 5 • 0 < 15? Test (0, 0) by substituting 0 for x and y. 0 – 0 < 15 0 < 15, true

5. 5 4 3 2 1 -3 -2 -1 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 Solution Step 3 If a true statement results, shade the half-plane containing the test point. Because 0 is less than 15, the test point (0, 0) is part of the solution set. All the points on the same side of the line 3x - 5y = 15 as the point (0, 0) are members of the solution set. The solution set is the half-plane that contains the point (0, 0), indicated by shading this half-plane. The graph is shown using green shading and a dashed blue line.