Question

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

Answer 1

Consider the line x+y=4

We observe that the shaded region and the origin are on the same sides of the line x+y=4 and (0,0) does not satisfy the inequation $x+y\le 4$

So,we must have one inequations as $x+y\le 4$

Consider the line y=3

We observe that the shaded region and the origin are on the same sides of the line y=3 and (0,0) does not satisfy the inequation $y\le 3$

So,the second inequations is $y\le 3$

Consider the line x=3

We observe that the shaded region and the origin are on the same sides of the line x=3 and (0,0) does not satisfy the inequation $x\le 3$

So,the third inequations is $x\le 3$

Consider the line x+5y=4

We observe that the shaded region and the origin are on the opposite sides of the line x+5y=4 and (0,0) does not satisfy the inequation $x+5y\ge 4$

Finaly,the fourth inequations is $x+5y\ge 4$

Consider the line 6x+2y=8

We observe that the shaded region and the origin are on the opposite sides of the line 6x+2y=8 and (0,0) does not satisfy the inequation 6x+2y=8

So,the fifth inequations is 6x+2y=8

We also notice that the shaded region is above x-axis and is on the right side of y-axis.So,we must have $x\ge 0$ and $y\ge 0$

Thus,the linear inequations corresponding to the given solution set are $x+y\le 4,y\le 3,x\le 3,x+5y\ge 4,6x+2y\ge 8,x\ge 0,y\ge 0$