Graphing Inequalities

To graph an inequality, treat the < , ≤ , > , or ≥ sign as an = sign, and graph the equation. If the inequality is < or > , graph the equation as a dotted line. If the inequality is ≤ or ≥ , graph the equation as a solid line. This line divides the xy - plane into two regions: a region that satisfies the inequality, and a region that does not.

Next, pick a point not on the line. Check to see if this point satisfies the inequality. If it satisfies the inequality, shade the region which contains that point. If it does not satisfy the inequality, shade the region which does not contain that point. All the points in the shaded region will satisfy the inequality.

Note: The origin (0, 0) is usually the easiest point to test, provided it is not on the line.

Example 1: Graph y≤2x - 3 .

Does (0, 0) satisfy y≤2x - 3 ? 0≤2(0) - 3 ? No. Shade the region that does not contain (0, 0):

Example 2: Graph 3x + 4y < 12 .

Does (0, 0) satisfy 3x + 4y < 12 ? 3(0) + 4(0) < 12 ? Yes. Shade the region that contains (0, 0):

Example 3: Graph y > x .

(0, 0) is on the line, so we must pick another point. Does (0, 1) satisfy y > x ? 1 > 0 ? Yes. Shade the region that contains (0, 1):

Example 4: Graph y≥x ² + 2 .

Does (0, 0) satisfy y≥x ² + 2 ? 0≥0² + 2 ? No. Shade the region that does not contain (0, 0):

To graph an inequality with a " ≠ " sign, change the ≠ sign to an = sign and graph the equation as a dotted line. Then shade both regions of the graph. The entire xy -graph, with the exception of the dotted line, satisfies the inequality.