# Graphing Linear Inequalities in Two Variables

To graph a linear
inequality
in two variables (say,
$x$
and
$y$
), first get
$y$
alone on one side. Then consider the
**
related equation
**
obtained by changing the inequality sign to an equals sign. The graph of this equation is a line.

If the inequality is
**
strict
**
(
$<$
or
$>$
), graph a dashed line. If the inequality is
**
not strict
**
(
**
$\le $
**
and
**
$\ge $
**
), graph a solid line.

Finally, pick one point not on the line ( $\left(0,0\right)$ is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

**
Example:
**

Graph the inequality $y\le 4x-2$ .

This line is already in
slope-intercept form
, with
$y$
alone on the left side. Its slope is
$4$
and its
$y$
-intercept is
$-2$
. So it's straightforward to graph it. In this case, we make a
**
solid
**
line since we have a "less than or equal to" inequality.

Now, substitute $x=0,y=0$ to decide whether $\left(0,0\right)$ satisfies the inequality.

$\begin{array}{l}0\stackrel{?}{\le}4\left(0\right)-2\\ 0\stackrel{?}{\le}-2\end{array}$

This is false. So, shade the half-plane which does
**
not
**
include the point
$\left(0,0\right)$
.