Beginning Algebra With Applications, Chapter 5, 5.6, Section 5.6, Problem 14

Illustrate the solution set $\displaystyle 3x + y \geq 6$


$
\begin{equation}
\begin{aligned}
3x + y &\geq 6
&& \text{Solve the inequality for } y \\
\\
y &\geq -3x + 6
&& \text{Subtract $-3x$ from each side}
\end{aligned}
\end{equation}
$

To graph the inequality, we first find the intercepts of the line $\displaystyle y = -3x + 6$.
In this case, the $x$-intercept (set $y = 0$) is $\left( 2, 0 \right)$


$
\begin{equation}
\begin{aligned}
0 &= -3x + 6 \\
\\
3x &= 6 \\
\\
x &= 2
\end{aligned}
\end{equation}
$


And the $y$-intercept (set $x = 0$) is $(0, 6)$

$
\begin{equation}
\begin{aligned}
y &=-3(0) + 6 \\
\\
y &= 6
\end{aligned}
\end{equation}
$

So the graph is


Graph is $y = -3x + 6 $ as a solid line. Shade the upper half of the plane.