Section 6.8 Linear Inequalities in Two Variables

What You Will Learn Graphing Linear Inequalities in Two Variables

Linear Inequality in Two Variables The solution set of a linear inequality in two variables is indicated on a coordinate plane. An inequality that is strictly less than (<) or greater than (>) will have as its solution set a half-plane. A half-plane is the set of all the points on one side of a line.

Linear Inequality in Two Variables An inequality that is less than or equal to (≤) or greater than or equal to (≥) will have as its solution set the set of points that consists of a half-plane and a line. To indicate that the line is part of the solution set, we draw a solid line. To indicate that the line is not part of the solution set, we draw a dashed line.

To Graph Linear Inequalities in Two Variables 1. Mentally substitute the equal sign for the inequality sign and plot points as if you were graphing the equation of a line. 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.

To Graph Linear Inequalities in Two Variables 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 the area on the same side of the line as the test point. If the test point results in a false statement, shade the area on the opposite side of the line as the test point.

Example 1: Graphing an Inequality Draw the graph of x + 2y < 4. Solution Draw the graph of x + 2y = 4 as a dashed line. Pick a test point not on the line (0, 0). 0 + 2(0) < 4 0 < 4 True

Example 1: Graphing an Inequality Solution

Example 2: Graphing an Inequality Draw the graph of 2x – 3y ≤ –6. Solution Draw the graph of 2x – 3y = –6 as a solid line. Pick a test point not on the line (0, 0). 2(0) – 3(0) ≤ –6 0 ≤ –6 False

Example 2: Graphing an Inequality Solution

Example 3: Graphing an Inequality Draw the graph of y < x. Solution Draw the graph of y = x as a dashed line. Pick a test point not on the line (1, –1). –1 < 1 True

Example 2: Graphing an Inequality Solution