Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 8

Illustrate the linear inequality $x + y \leq -3$ in two variables.

To graph $x + y \leq -3$ we must graph the boundary line $x + y = -3$ first. To do this, we need to find the
intercepts of the line

$x$-intercept (set $y = 0$):

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


$y$-intercept (set $x = 0$):

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


Now, by using test point. Let's say point $(-4,-2)$ from the left of the boundary line.

$
\begin{equation}
\begin{aligned}
x + y &\leq - 3\\
\\
-4 + (-2) &\leq -3 \\
\\
-4 - 2 &\leq -3\\
\\
-6 &\leq -3
\end{aligned}
\end{equation}
$


Since the inequality symbol is $\leq$, then the boundary line must be solid.
Moreover, since the test point satisfy the inequality, then we must shade the left
portion of the boundary line. So the graph is,