Step 1:

The first inequality is $3x +2y \leq 150$------(1)

Consider the equation $3x +2y =150$

The points $(50,0) $ and $(0,75)$ satisfy the equation .

The graph of the equation is drawn as shown.

The line divides the xy plane into two half planes.

Consider the point $(0,0)$

We see that,

$3(0) +2(0) \leq 150$

=> $ 0 \leq 150$ is true.

Thus the inequality (1) is represented by the region below the line $ 3x+2y =150$ Containing the point $(0,0) $ (including the line)

Step 2:

The second inequality is $ x+4y \leq 80$

Consider the equation $x+4y=80$

The point $(80,0) $ and $(0,20)$ satisfy the equation.

The graph of the equation is drawn as shown.

The line divides the xy plane into two half parts.

Consider the point (0,0)

We see that $)+ 4(0) \leq 80$

=> $0 \leq 8$ is true.

Thus the inequality (2) is represented by the region below the line $ x+4y =80$ Containing the point $(0,0) $ (including the line)

Step 3:

The third is $ x \leq 15$

The inequality (3) is represented by the region to the lest of the line $ x =15$

Step 4:

The fourth inequality $ x \leq 0; y \leq 0$ represents the first quadrant.

Step 5:

Hence the solution of the system of linear inequalities is represented by the common shaded region in the first quadrant including the corresponding lines.