Answer
Please see the graph.
Work Step by Step
Blue line: $5x-3y\le12$
Orange line: $y+2x\ge0$
Purple line: $y\le2$
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:
$5x-3y\le12$
$5*0-3*0\le12$
$0\le12$ (true, so we shade above the line)
Orange line:
$y+2x\ge0$
$0+2*0\ge0$
$0\ge0$ (true, so we shade the right side of the line)
Purple line:
$y\le2$
$0\le2$ (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 $(-1,2)$, $(3.6, 2)$, and $(1.091, -2.182)$.
