Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 10

Illustrate the linear inequality $3x - y < 3$ in two variables.

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

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

$
\begin{equation}
\begin{aligned}
3x - 0 &= 3 \\
\\
3x &= 3 \\
\\
x &= 1
\end{aligned}
\end{equation}
$


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

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


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

$
\begin{equation}
\begin{aligned}
3x - y &< 3\\
\\
3(0) - 0 &< 3\\
\\
0 &< 3
\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 left
portion of the boundary line. So the graph is,