1. Section 6.8 Linear Inequalities in Two Variables

2. What You Will Learn
Graphing Linear Inequalities in Two Variables

3. 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.

4. 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.

5. 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.

6. 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.

7. 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

8. Example 1: Graphing an Inequality
Solution

9. 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

10. Example 2: Graphing an Inequality
Solution

11. 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

12. Example 2: Graphing an Inequality
Solution