Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 16

Illustrate the linear inequality $4x - 5y > 20$ in two variables.

To graph $4x - 5y > 20$ we must graph the boundary line $4x - 5y = 20$ first. To do this, we need to find the
intercepts of the line

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

$
\begin{equation}
\begin{aligned}
4x - 5(0) &= 20 \\
\\
4x &= 20 \\
\\
x &= 5
\end{aligned}
\end{equation}
$


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

$
\begin{equation}
\begin{aligned}
4(0) - 5y &= 20 \\
\\
-5y &= 20 \\
\\
y &= - 4
\end{aligned}
\end{equation}
$


Now, by using test point. Let's say point $(4,-3)$ from the right of the boundary line.

$
\begin{equation}
\begin{aligned}
4x - 5y &> 20\\
\\
4(4) -5(-3) &> 20\\
\\
16 + 15 &> 20 \\
\\
31 &> 20
\end{aligned}
\end{equation}
$


Since the inequality symbol is $ > $, then the boundary line must be dashed.
Moreover, since the test point satisfy the inequality, then we must shade the right
portion of the boundary line. So the graph is,