Beginning Algebra With Applications, Chapter 5, Cumulative Exercises, Section Cumulative Exercises, Problem 18

Illustrate the solution set of $x - y \leq 5$

$
\begin{equation}
\begin{aligned}
x -y &\leq 5\\
\\
-y &\leq -x + 5
&& \text{Solve the inequality for } y \\
\\
\frac{-y}{-1} &\geq \frac{-x}{-1} + \frac{5}{-1}
&& \text{Remember that if you divide or multiply negative numbers ,the inequality symbol reverses}\\
\\
y &\geq x - 5
\end{aligned}
\end{equation}
$


To graph the inequality, we first find the intercepts of the line $y = x - 5$
In this case, the $x$-intercept (set $y = 0$) is $(5,0)$

$
\begin{equation}
\begin{aligned}
0 &= x - 5\\
\\
x &= 5
\end{aligned}
\end{equation}
$

And the $y$-intercept (set $x = 0$) is $(0, -5)$

$
\begin{equation}
\begin{aligned}
y &= 0 -5 \\
\\
y &= -5
\end{aligned}
\end{equation}
$

So the graph is


Graph $y = x - 5$ as a solid line. Shade the upper half of the plane.