Solving Systems of Linear Inequalities

The solution of a system of linear inequalities in two variables is the set of ordered pairs that satisfy all the inequalities of the system. The solution can be found by graphing each inequality and noting where the graphs overlap.

 

Procedure — To Solve a System of Two Linear Inequalities in Two Variables

Step 1 Solve the first inequality for y. Then graph the inequality.

Step 2 Solve the second inequality for y. Then graph the inequality.

Step 3 Shade the region where the two graphs overlap.

The solution is the set of all ordered pairs in the shaded region.

To graph each inequality, first write its corresponding equation in slopeintercept form, y = mx + b.

When graphing each inequality, use either a solid line or a dotted line, as follows:

• If the original inequality symbol is ≤ or ≥, use a solid line to show that points on the line are solutions of the inequality.

• If the original inequality symbol is < or >, use a dotted line to show that points on the line are not solutions of the inequality.

The region that satisfies each inequality can be identified by using the following guidelines:

Note:

Before you use these guidelines, be sure that each inequality has been solved for y. That is, be sure that y is by itself on the left side of each inequality.