Each line plotted on a coordinate graph divides the graph (or plane) into two *half‐planes.* This line is called the *boundary line* (or *bounding line*). The graph of a linear inequality is always a half‐plane. Before graphing a linear inequality, you must first find or use the equation of the line to make a boundary line.

Open half-plane

If the inequality is a “>” or “<”, then the graph will be an *open half‐plane.* An open half‐plane does not include the boundary line, so the boundary line is written as a *dashed line* on the graph.

Example 1

Graph the inequality *y* < *x* – 3.

First graph the line *y* = *x* – 3 to find the boundary line (use a dashed line, since the inequality is “<”) as shown in Figure 1.

Figure 1. Graph of boundary line for *y* < *x* – 3.

x

y

| 3 | 0

| 0 | -3

| 4 | 1

Now shade the lower half‐plane as shown in Figure 2, since *y* < *x* – 3.

Figure 2. Graph of inequality *y* < *x* – 3.

To check to see whether you've shaded the correct half‐plane, plug in a pair of coordinates—the pair of (0, 0) is often a good choice. If the coordinates you selected make the *inequality a true statement* when plugged in, then you *should* shade the half‐plane *containing* those coordinates. If the coordinates you selected *do not* make the inequality a true statement, then shade the half‐plane *not containing* those coordinates.

Since the point (0, 0) *does not* make this inequality a true statement,

*y* < *x* – 3

0 < 0 – 3 is not true.

You should shade the side that *does not contain* the point (0, 0).

This checking method is often simply used as the method to decide which half‐plane to shade.

Closed half-plane

If the inequality is a “≤”or “≥”, then the graph will be a *closed half‐plane*. A closed half‐plane includes the boundary line and is graphed using a *solid line and shading.*

Example 2

Graph the inequality 2 *x* – *y* ≤ 0.

First transform the inequality so that *y* is the left member.

Subtracting 2 *x* from each side gives

– *y* ≤ –2 *x*

Now dividing each side by –1 (and changing the direction of the inequality) gives

*y* ≥ 2 *x*

Graph *y* = 2 *x* to find the boundary (use a solid line, because the inequality is “≥”) as shown in Figure 3.

Figure 3. Graph of the boundary line for y ≥ 2x.

x

y

| 0 | 0

| 1 | 2

| 2 | 4

Since *y* ≥ 2 *x*, you should shade the upper half‐plane. If in doubt, or to check, plug in a pair of coordinates. Try the pair (1, 1).

So you should shade the half‐plane that *does not contain* (1, 1) as shown in Figure 4.

Figure 4. Graph of inequality *y* ≥ 2 *x*.