Graphing Linear Inequalities
Use the following procedure to graph a linear inequality in two variables.
Procedure â€”
To Graph a Linear Inequality in Two Variables
Step 1 Graph the equation that corresponds to the given inequality.
â€¢ If the inequality symbol is ≤ or
≥, use a solid line to show that
points on the line are solutions of the inequality.
â€¢ If the inequality symbol is < or >, use a dotted line to show
that points on the line are not solutions of the inequality.
Step 2 Use a test point NOT on the line to determine the region
whose points satisfy the inequality.
Step 3 Shade the region whose points satisfy the inequality.
Example 1
Graph the inequality 2x  y < 4.
Solution
Step 1 Graph the equation that corresponds to the given inequality.
Graph the equation 2x  y = 4.
To do this, substitute 0 for y and solve for x to get the xintercept, (2, 0).
Next, substitute 0 for x and solve for y to get the yintercept, (0, 4).
Then, plot the points.
Since the inequality symbol â€œ<â€ does not contain â€œequal to,â€ draw a
dotted line through the plotted points.
The dotted line shows that points on the line are not 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 and 0 for y.
Simplify. 
Is Is 
2(0)  0 0 
< 4 ?
< 4 ? Yes 
Since 0 < 4 is true, the ordered pair (0, 0)
is a solution of the inequality 2x  y < 4.
This means all the points in the region containing (0, 0) are solutions.
Step 3 Shade the region whose points satisfy the inequality.
Shade the region that includes (0, 0). This is the region above the dotted
line.
Note â€” Using (0, 0) as a Test Point
If the point (0, 0) does not lie on the line, it is a good test point since
it often makes the calculations easier.
