The solutions of an inequality in two variables are represented on a graph by shading the region above or below the graph of the related equation.

The set of solutions, or solution set, of an inequality in one variable, such as $x>3$, can be graphed on a number line. The solution set of an inequality in two variables, such as $y>x$, is represented by a shaded region in the plane, with a boundary line formed by the related equation. The related equation is formed by replacing the inequality symbol with an equal sign.

To graph the solutions of a linear inequality in two variables:

1. Solve the inequality for $y$, if needed. The inequality should be written in one of these forms:$\begin{aligned}y&<mx+b\\y&\leq mx+b\\y&>mx+b\\y&\geq mx+b\end{aligned}$

- If the inequality symbol is $<$ or $>$, use a dashed line to show that the boundary line is not part of the solution set.
- If the inequality symbol is $\leq$ or $\geq$, use a solid line to show that the boundary line is included in the solution set.

3. Shade the side of the line that represents the solution set.

- If the inequality symbol is $<$ or $\leq$, shade below the line to show the $y$-values that are less than the $y$-values on the line.
- If the inequality symbol is $>$ or $\geq$, shade above the line to show the $y$-values that are greater than the $y$-values on the line.
- A test point may be used to check whether the correct side of the line is shaded. Choose a point not on the line, and substitute the $x$- and $y$-coordinates into the inequality. If the result is true, the side of the line containing that point should be shaded.

### Graphs of Inequalities in Two Variables

Step-By-Step Example

Graphing a Linear Inequality

Graph the linear inequality:

$y-3x\gt-2$

Step 1

Solve for $y$ by adding $3x$ to both sides.

$\begin{aligned}y-3x &> -2\\y&>3x-2\end{aligned}$

Step 2

Graph the boundary line:
Use a dashed line because the symbol is greater than.

$y=3x-2$

Solution

The inequality symbol is greater than. So shade above the boundary line.
To check that the correct side of the line is shaded, select a test point that is not on the boundary line, and substitute the coordinates into the original inequality. The point $(0,0)$ is not on the line. It is in the shaded region, so the inequality should be true.
The inequality is true. So, the region that contains $(0,0)$ is the solution.

$\begin{aligned}y-3x&>-2\\(0)-3(0)&\stackrel{?}{>}-2 \\0-0&\stackrel{?}{>}-2\\0&>-2&\space\checkmark\end{aligned}$