# Graphing Inequalities in Two Variables

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&\geq mx+b\end{aligned}
2. Graph the boundary line, $y=mx+b$ .
• 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

 The inequality symbol is $\lt$ , so the boundary line is dashed. The values of $y$ are less than the values of $x$ , so the points below the line $y=x$ are shaded. The inequality symbol is $>$ , so the boundary line is dashed. The values of $y$ are greater than the values of $x$ , so the points above the line $y=x$ are shaded. The inequality symbol is $\leq$ , so the boundary line is solid. The values of $y$ are less than the values of $x$ , so the points below the line $y=x$ are shaded. The inequality symbol is $\geq$ , so the boundary line is solid. The values of $y$ are greater than the values of $x$ , so the points above the line $y=x$ are shaded.

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:
$y=3x-2$
Use a dashed line because the symbol is greater than.
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.
\begin{aligned}y-3x&>-2\\(0)-3(0)&\stackrel{?}{>}-2 \\0-0&\stackrel{?}{>}-2\\0&>-2&\space\checkmark\end{aligned}
The inequality is true. So, the region that contains $(0,0)$ is the solution.