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.