Solving Systems of Linear Inequalities Graphically

In order for $(1,4)$ to be a solution to the system of inequalities, we have to shade the region to the right of both inequalities. Let's do that and see what happens.

Let's isolate the shading where both inequalities apply.

Thus, this is the correct area since $(1,4)$ is in the shaded area but not the other three points. We should now find the correct inequality signs buy substituting $(1,4)$ into the boundary lines. In order to form a true statement we need to use $>$ or $\ge .$ Since the line is solid, we should use $\ge .$ $$y\ge \text{-}3x+2$$ Nwo we do the same thing with the second line. For the second inequality we can use $\ge$ for it to be true. To summarize the inequalities should be

$${\displaystyle \begin{cases}y \ \fbox{\geq} \ \text{-} 3x+2 \\ y \ \fbox{\geq} \ x+2 \end{cases} }$$