Question

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

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 \[\leq\] 4Considering 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\[\leq\]3 So, the corresponding inequation is y\[\leq\]3Considering 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\[\leq\] 3 So, the corresponding inequation is x  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\[\geq 4\]So, the corresponding inequation is x + 5y \[\geq 4\] 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\[\geq 8\]So, the corresponding inequation is 6x + 2y\[\geq 8\]Also the shaded region is in the first quadrant. Therefore, we must have \[x \geq 0 \text{ and } y \geq 0\]

Thus, the linear inequations comprising the given solution set are given below:
  x + y\[\leq\]4, y\[\leq\]3, x\[\leq\]3, x + 5y\[\geq 4\]6x + 2y\[\geq 8\]\[x \geq 0 \text{ and } y \geq 0\]