Answer
Please see the graph.
Work Step by Step
Blue line: $5x+3y\ge-2$
Orange line: $y+2x\leq0$
Purple line: $y\leq4$
To determine what sides of the lines to shade, we use a point not on the line. We can use (0,0) for the lines.
Blue line:
$5x+3y\ge-2$
$5*0+3*0\ge-2$
$0\ge-2$ (true, so we shade the right side of the line)
Orange line:
$y+2x\leq0$
$0+2*0\leq0$
$0\leq0$ (true, so we shade the left side of the line)
Purple line:
$y\leq4$
$0\leq4$ (true, so we shade below the line)
The solution area is the overlap of the three inequalities. In this case, the area is the triangle created at the three intersection points of $(-2.8,4)$, $(-2,4)$, and $(2, -4)$.