Graphing Linear Inequalities
Example 1
Graph the inequality x ≥ 3.
Solution
Step 1 Graph the equation that corresponds to the given inequality.
Graph the equation x = 3.
The graph is a vertical line through (3, 0).
Since the inequality symbol â€œ≥â€ contains â€œequal to,â€ draw a solid line.
The solid line shows that points on the line are solutions of the
inequality.
Step 2 Use a test point NOT on the line to determine the region whose
points satisfy the inequality.
The point (0, 0) is not on the line, so it can be used as a test point.
Substitute 0 for x. The resulting statement, 0
≥ 3, is false.
Therefore, the solutions do not lie in the region containing (0, 0).
The solutions lie in the other region and on the line.
Step 3 Shade the region whose points satisfy the inequality.
Since the test point (0, 0) does NOT satisfy x
≥ 3, shade the region that does NOT include (0, 0).
This is the region to the right of the solid line, including the solid line.
Note:
A vertical line divides the xyplane into
three regions:
â€¢ points on the line
â€¢ points to the right of the line
â€¢ points to the left of the line.
Example 2
Graph the inequality y ≥ x.
Solution
Step 1 Graph the equation that corresponds to the given inequality.
The equation y = x can be written as y = 1x + 0. The yintercept
is (0, 0) and the slope is 1. We use this information to graph the
equation.
Since the inequality symbol â€œ≥â€ contains â€œequal to,â€ draw a solid line to
show that points on the line are solutions of the inequality.
Step 2 Use a test point NOT on the line to determine the region whose
points satisfy the inequality.
The point (0, 0) is on the line, so we must select a different test point.
Letâ€™s use (2, 0).
In y ≥ x, substitute 2 for x and 0 for y. The resulting statement,
0 ≥ 2, is true.
Therefore, the solutions lie in the region that contains (2, 0).
Step 3 Shade the region whose points satisfy the inequality.
Since the test point (2, 0) satisfies y ≥ x, shade the region that contains (2, 0).
This is the region above the solid line, including the solid line.
