Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 18

Illustrate the linear inequality $x + 2y > 0$ in two variables.

To graph $x + 2y > 0$ we must graph the boundary line $x + 2y = 0$ first. To do this, we need to find the
intercepts of the line

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

$
\begin{equation}
\begin{aligned}
x + 2(0) &= 0 \\
\\
x &= 0
\end{aligned}
\end{equation}
$


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

$
\begin{equation}
\begin{aligned}
(0) + 2y &= 0 \\
\\
2y &= 0 \\
\\
y &= 0
\end{aligned}
\end{equation}
$


Now, by using test point. Let's say point $(2,2)$ from the right of the boundary line.

$
\begin{equation}
\begin{aligned}
x + 2y &> 0 \\
\\
2 + 2(2) &> 0 \\
\\
2 + 4 &> 0 \\
\\
6 &> 0
\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,