**Question:**

Find the linear inequations for which the shaded area in Fig. 15.41 is the solution set. Draw the diagram of the solution set of the linear inequations:

**Solution:**

Considering the line $2 x+3 y=6$, we find that the shaded region and the origin $(0,0)$ are on the opposite side of this line and $(0,0)$ does not satisfy the inequation $2 x+3 y \geq 6$ So, the first inequation is $2 x+3 y \geq 6$

Considering the line $4 x+6 y=24$, we find that the shaded region and the origin $(0,0)$ are on the same side of this line and $(0,0)$ satisfies the inequation $4 x+6 y \leq 24$ So, the corresponding inequation is $4 x+6 y \leq 24$

Considering the line $x-2 y=2$, we find that the shaded region and the origin $(0,0)$ are on the same side of this line and $(0,0)$ satisfies the inequation $x-2 y \leq 2$ So, the corresponding inequation is $x-2 y \leq 2$

Considering the line $-3 x+2 y=3$, we find that the shaded region and the origin $(0,0)$ are on the same side of this line and $(0,0)$ satisfies the inequation $-3 x+2 y \leq 3$ So, the corresponding inequation is $-3 x+2 y \leq 3$

Also, the shaded region is in the first quadrant. Therefore, we must have $x \geq 0$ and $y \geq 0$

Thus, the linear inequations comprising the given solution set are given below:

$2 x+3 y \geq 6,4 x+6 y \leq 24, x-2 y \leq 2,-3 x+2 y \leq 3, x \geq 0$ and $y \geq 0$