Beginning Algebra With Applications, Chapter 5, 5.6, Section 5.6, Problem 12

Illustrate the solution set $\displaystyle x - y > - 3$


$\begin{equation} \begin{aligned} x - y &> - 3 && \text{Solve the inequality for } y\\ \\ -y &> -x - 3\\ \\ \frac{y}{-1} &< \frac{-x}{-1} - \frac{3}{-1} && \text{Remember that if you divide or multiply negative numbers ,the inequality symbol reverses}\\ \\ y &< x + 3 \end{aligned} \end{equation}$

To graph the inequality, we first find the intercepts of the line $\displaystyle y = x + 3$.
In this case, the $x$-intercept (set $y = 0$) is $\left( -3, 0 \right)$


$\begin{equation} \begin{aligned} 0 &= x + 3\\ \\ x &= -3 \end{aligned} \end{equation}$


And the $y$-intercept (set $x = 0$) is $(0, 3)$

$\begin{equation} \begin{aligned} y &= 0 + 3\\ \\ y &= 3 \end{aligned} \end{equation}$

So the graph is


Graph is $y = x + 3$ as a dashed line. Shade the lower half-plane.