Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 20

Illustrate the linear inequality $x - 5y \leq 0 $ in two variables.
To graph $x - 5y \leq 0$ we must graph the boundary line $x - 5y = 0$ first. To do this, we need to find the
intercepts of the line

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

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


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

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


The $x$ and $y$-intercepts are the origin or at point $(0,0)$

Now, by using test point, let's say point $(-6,1)$ for the left of the boundary line,

$
\begin{equation}
\begin{aligned}
x - 5y &\leq 0 \\
\\
-6 - 5 (1) &\leq 0 \\
\\
-6 - 5 &\leq 0 \\
\\
-11 &\leq0
\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,