Beginning Algebra With Applications, Chapter 5, 5.6, Section 5.6, Problem 18

Illustrate the inequality $-4x + 3y < - 12$

$
\begin{equation}
\begin{aligned}
3y &< 4x - 12
&& \text{Solve the inequality for } y \\
\\
y &< \frac{4}{3}x - 4
\end{aligned}
\end{equation}
$

To graph the inequality, we first find the intercepts of the line $\displaystyle y = \frac{4}{3}x - 4$.
In this case, the $x$-intercept (set $y = 0$) is $(3,0)$

$
\begin{equation}
\begin{aligned}
0 &= \frac{4}{3}x - 4\\
\\
\frac{4}{3}x &= 4 \\
\\
x &= 3
\end{aligned}
\end{equation}
$


And the $y$-intercept (set $x = 0$) is $(0, -4)$

$
\begin{equation}
\begin{aligned}
y &= \frac{4}{3} (0) - 4\\
\\
y &= -4
\end{aligned}
\end{equation}
$


So the graph is



Graph $\displaystyle y = \frac{4}{3}x - 4$ as a dashed line. Shade the lower half of the plane.