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

First consider the linear equation x + y = 8, Observe that the shaded region and the origin both are on the same side of the graph of the line and (0, 0) satisfy the constraint x + y ≤ 8.

Now, consider x + y = 4, observe that the origin is on the opposite side of the shaded region and (0, 0) doesn’t satisfy the constraint x + y ≥ 4, therefore required constraint is x + y ≥ 4

Also, shaded region is in the first quadrant i.e. x ≥ 0 and y ≥ 0,

Also, shades region is below the line y = 5 and left to the line x = 5

⇒ y ≤ 5 and x ≥ 5

Hence, the linear inequalities are

x + y ≤ 8

x + y ≥ 4

x ≥ 0

y ≥ 0

x ≤ 5

y ≤ 5