Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 22

Illustrate the linear inequality $y \leq 4x$ in two variables.
To graph $y \leq 4x$ we must graph the boundary line $y = 4x$ first. To do this, we need to find the
intercepts of the line

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

$
\begin{equation}
\begin{aligned}
0 &= 4x\\
\\
0 &= x
\end{aligned}
\end{equation}
$


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

$
\begin{equation}
\begin{aligned}
y &= 4(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 $(1,1)$ from the right of the boundary line.

$
\begin{equation}
\begin{aligned}
y &\leq 4x\\
\\
1 &\leq 4(1)\\
\\
1 &\leq 4
\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 right
portion of the boundary line. So the graph is,