Answer
Please see the graph.
Work Step by Step
Green line: $4x-y\ge-2$
Blue line: $2x+3y\le-8$
Orange line: $y\ge-5$
Let's use the point $(0,0)$ to determine what sides of the lines to shade.
Green line: $4x-y\ge-2$
$4x-y\ge-2$
$4*0-0\ge -2$
$0-0\ge -2$
$0 \ge -2$ (true)
This is true, so we shade the side of the line that does have the point. The inequality has the "greater than or equal to" sign, so we use a solid line.
Blue line: $2x+3y\le-8$
$2x+3y\le-8$
$2*0+3*0\le-8$
$0+0 \le -8$
$0 \le -8$ (false)
This is false, so we shade the side of the line that doesn't have the point. The inequality has the "less than or equal to" sign, so we use a solid line.
Orange line: $y\ge-5$
$y\ge-5$
$0 \ge -5$ (true)
This is true, so we shade the side of the line that does have the point. The inequality has the "greater than or equal" sign, so we use a solid line.
The solution area for the system on inequalities is the area that is shaded blue, green, and orange.