Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 12

Illustrate the linear inequality $x + 4y \geq - 3$ in two variables.

To graph $x + 4y \geq -3$ we must graph the boundary line $x + 4y = - 3$ first. To do this, we need to find the
intercepts of the line

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

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


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

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


Now, by using test point. Let's say point $(0,0)$ from the right of the boundary line.

$
\begin{equation}
\begin{aligned}
x + 4y &\geq - 3\\
\\
0 + 4(0) &\geq -3 \\
\\
0 &\geq -3
\end{aligned}
\end{equation}
$


Since the inequality symbol is $ \geq $, 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,