Answer
Please see the graph.
Work Step by Step
Green line: $y+2x\le0$
Blue line: $5x-3y\ge-2$
Orange line: $y\le4$
Let's use the point $(0,0)$ to determine what sides of the lines to shade.
Green line: $y+2x\le0$
$y+2x\le0$
$0+2*0\le 0$
$0+0\le 0$
$0 \le 0$ (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.
Blue line: $5x-3y\ge-2$
$5x-3y\ge-2$
$5*0-3*0\ge-2$
$0-0 \ge -2$
$0 \ge -2$ (true)
This is true, so we shade the side of the line that does have the point. The inequality has the "less than" sign, so we use a dotted line.
Orange line: $y\le4$
$y\le4$
$0 \le 4$ (true)
This is true, so we shade the side of the line that does have the point. The inequality has the "less 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.