Solving Systems of Linear Inequalities

Example

Solution

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

The first inequality does not contain the variable y.

To graph the first inequality, x ≤ 0, first graph the corresponding equation, x = 0.

• This is a vertical line that passes through the x-axis at the point (0, 0); it is the y-axis.

For the inequality x ≤ 0, the inequality symbol is “≤”. This stands for “is less than or equal to.”

• To represent “less than,” shade the region to the left of the line. Each point in that region has an x-coordinate less than 0.

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

To solve x - 2y > 2 for y, do the following:

Subtract x from both sides.	- 2y > - x + 2
Divide both sides by -2. Be sure to reverse the inequality symbol because you are dividing by a negative number.
Simplify.

• The y-intercept is (0, -1). Plot (0, -1).

• The slope is . To find a second point on the line, start at (0, -1) and move up 1 and right 2 to the point (2, 0). Plot (2, 0).

• Since the inequality symbol “<” does not contain “equal to,” draw a dotted line through (0, -1) and (2, 0).

• To represent “less than,” shade the region below the line.

Step 3 Shade the region where the two graphs overlap.

The solution is the region where the graphs overlap.

The solution of the system is the dark shaded region.

As a check, choose a point in the solution region.

For example, choose ( -1, -5).

To confirm that ( -1, -5) is a solution of the system, substitute -1 for x and -5 for y in each of the original inequalities and simplify.

Since ( 1, 5) satisfies each inequality, it is a solution of the system.