1. Inequalities and Absolute Value8
Chapter
Inequalities and Absolute Value

2. Linear Inequalities and Systems in Two Variables
8.4
Linear Inequalities and Systems in Two Variables
1. Graph linear inequalities in two variables. 2. Graph systems of linear inequalities by graphing.

3. Graph linear inequalities in two variables.
Objective 1
Graph linear inequalities in two variables.

4. Graph linear inequalities.
In our early work, we graphed linear inequalities in one variable on a number line. In this section, we graph linear inequalities in two variables on a rectangular coordinate system.

5. Graph linear inequalities.
Consider the graph. The graph of the line x + y = 5 divides the points in the rectangular coordinate system into three sets: 1. Those points that lie on the line itself and satisfy the equation x + y = 5, such as (0, 5), (2, 3), and (5, 0); 2. Those points that lie in the region above the line and satisfy the inequality x + y > 5, such as (5, 3) and (2, 4); 3. Those points that lie in the region below the line and satisfy the inequality x + y < 5, such as (0, 0) and (–3, –1).

6. Graphing a Linear Inequality in Two Variables
Step 1 Draw the graph of the straight line that is the boundary. ● Make the line solid if the inequality involves ≤ or ≥ . ● Make the line dashed if the inequality involves < or >. Step 2 Choose a test point. Choose any point not on the line, and substitute the coordinates of that point in the inequality. Step 3 Shade the appropriate region. Shade the region that includes the test point if it satisfies the original inequality. Otherwise, shade the region on the other side of the boundary line.

7. Graphing a Linear Inequality
Classroom Example 1
Graphing a Linear Inequality
Graph x + y ≤ 4. Graph the boundary line x + y = 4, which has intercepts (0, 4) and (4, 0). Select a test point: (0, 0). Shade the region containing the test point, since it was a true statement.

8. Graph linear inequalities.
If the inequality is written in the form y > mx + b or y < mx + b, then the inequality symbol indicates which half-plane to shade. If y > mx + b, then shade above the boundary line. If y < mx + b, then shade below the boundary line. This method works only if the inequality is solved for y.

9. Graphing a Linear Inequality with Boundary Passing through the Origin
Classroom Example 2
Graphing a Linear Inequality with Boundary Passing through the Origin
Graph x + y > 0. Graph the equation. Solve the inequality for y. y > –x Shade the region above the boundary line.

10. Graphing a Linear Inequality
Classroom Example 3
Graphing a Linear Inequality
Graph y + 4 ≤ 0. Graph the equation y = –4. Choose a test point: (0, 0) ≤ 0 4 ≤ 0 FALSE Shade the region that does not contain the test point.

11. Solve systems of linear inequalities by graphing.
Objective 2
Solve systems of linear inequalities by graphing.

12. Solving Systems of Linear Inequalities by Graphing
A system of linear inequalities consists of two or more linear inequalities. The solution set of a system of linear inequalities includes all ordered pairs that make all inequalities of the system true at the same time.

13. Solving a System of Linear Inequalities
Step 1 Graph each inequality. Use the method of Objective 1. Step 2 Choose the intersection. Indicate the solution set by shading the intersection of the graphs—that is, the region where the graphs overlap.

14. Solving a System of Linear Inequalities
Classroom Example 4
Solving a System of Linear Inequalities
Graph the solution set of the system. Graph the first inequality. Graph the second inequality. The solution set of this system includes all points in the intersection—that is, the overlap—of the graphs of the two inequalities.
solution set

15. Solving a System of Linear Inequalities
Classroom Example 5
Solving a System of Linear Inequalities
Graph the solution set of the system. Use dashed lines and graph each inequality. The solution set of the system is the region with light purple shading. Neither boundary line is included.
solution set