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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*≤2*x* - 3.

Step 1 -- Graph of *y* = 2*x* - 3

Does (0, 0) satisfy

*y*≤2*x* - 3 ?

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

*y*≤2*x* - 3

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

**Step 1 -- Graph of 3***x* + 4*y* = 12
Does (0, 0) satisfy

3*x* + 4*y* < 12 ?

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

3*x* + 4*y* < 12

*Example 3*: Graph *y* > *x*.

Step 1--Graph of *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):

*y* > *x*