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.