Graph each system of linear inequalities by hand. Verify your results using a graphing utility.

$$\begin{gathered} 3x-y\ge 6 \\ x+2y\le 2 \end{gathered}$$

We are given a system of linear inequalities $$\begin{gathered} 3x-y\ge 6 \\ x+2y\le 2 \end{gathered}$$and we need to find the solution region of the inequalities.

First, we plot the equation of the line $3x-y=6$.

The graph of the inequality $3x-y\ge 6$is a solid line and so is part of the graph of the inequality. Since $\left(0,0\right)$does not satisfy the inequality, the graph consists of the portion away from the origin. The shaded portion below the line represents the inequality. The graph is

Again, we plot the equation of the line $x+2y=2$.

The graph of the inequality $x+2y\le 2$is a solid line and so is part of the graph of the inequality. Since $\left(0,0\right)$satisfies the inequality, the graph consists of the portion towards the origin. The shaded portion below the line represents the inequality. The graph is

The intersection of the shaded regions of the two inequalities gives the result. The required solution is