Question

Linear inequalities for which the shaded region for the given figure is the solution set, are

• A. $x+y \leq 8, x+y \leq 4, x \leq 5, y \leq 5, x \geq 0, y \geq 0$
• B. $x+y \leq 8, x+y \geq 4, x \leq 5, y \leq 5, x \geq 0, y \geq 0$
• C. $x+y \geq 8, x+y \geq 4, x \geq 5, y \geq 5, x \geq 0, y \geq 0$
• D. None of the above

Answer 1

Answer: (B)

(i) Consider the line $x+y=8$. We observe that the shaded region and origin lie on the same side of this line and $(0,0)$ satisfies $x+y \leq 8$. Therefore, $x+y \leq 8$ is the linear inequality corresponding to the line $x+y=8$.
(ii) Consider $x+y=4$. We observe that shaded region and origin are on the opposite side of this line and $(0,0)$ satisfies $x+y \leq 4$. Therefore, we must have $x+y \geq 4$ as linear inequalities corresponding to the line $x+y=4$
(iii) Shaded portion lie below the line $y=5$. So, $y \leq 5$ is the linear inequality corresponding to $y=5$.
(iv) Shaded portion lie on the left side of the line $x=5$. So, $x \leq 5$ is the linear inequality corresponding to $x=5$.
(v) Also, the shaded region lies in the first quadrant only. Therefore, $x \geq 0, y \geq 0$.
In view of (i), (ii), (iii), (iv) and (v) above the linear inequalities corresponding to the given solutions are $x+y \leq 8, x+y \geq 4, y \leq 5, x \leq 5, x \geq 0$ and $y \geq 0$