Question

Find the linear inequations for which the solution set is the shaded region given in Fig. 15.42

Figure

Answer 1

Considering the line x + y = 4, we find that the shaded region and the origin (0, 0) are on the same side of this line and (0, 0) does not satisfy the inequation x + y $\le$ 4 So, the first inequation is x + y $\le$ 4

Considering the line 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 y $\le$ 3 So, the corresponding inequation is y $\le$ 3

Considering the line x = 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 x $\le$ 3 So, the corresponding inequation is x $\le$ 3

Considering the line x + 5y = 4, 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 x + 5y $\ge 4$ So, the corresponding inequation is x + 5y $\ge 4$

Considering the line 6x + 2y = 8, 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 6x + 2y $\ge 8$ So, the corresponding inequation is 6x + 2y $\ge 8$

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

Thus, the linear inequations comprising the given solution set are given below:
x + y $\le$ 4, y $\le$ 3, x $\le$ 3, x + 5y $\ge 4$, 6x + 2y $\ge 8$, $x\ge 0\mathrm{and}y\ge 0$