The linear inequalities satisfying the shaded feasible region given in the figure are

(A) $x \geq 0$, $y \geq 0$, $2x + y \geq 2$

(B) $x \geq 0$, $y \geq 0$, $2x + y \leq 2$

(C) $x \geq 0$, $y \geq 0$, $2x + y \geq 2$, $x + 2y \leq 8$, $x - y \leq 1$

(D) $x + 2y \geq 8$, $x - y \geq 1$

Choose the correct answer from the options given below:

The feasible region shown in the graph needs to be described by a set of linear inequalities.

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From the graph, the region is in the first quadrant:

$x \geq 0$ and $y \geq 0$

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The boundary lines forming the shaded region are:

Line passing through $(1, 0)$ and $(0, 2)$ has equation $2x + y = 2$

At $(1, 0)$: $2(1) + 0 = 2$ ✓

The shaded region is above this line: $2x + y \geq 2$

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Another boundary line has equation $x + 2y = 8$

The region is below this line: $x + 2y \leq 8$

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Another boundary line has equation $x - y = 1$

The region is below this line: $x - y \leq 1$

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Option (A): $x \geq 0$, $y \geq 0$, $2x + y \geq 2$

This describes an unbounded region in the first quadrant above the line $2x + y = 2$. The shaded region is part of this larger region. This is correct.

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Option (B): $x \geq 0$, $y \geq 0$, $2x + y \leq 2$

This describes a region below the line $2x + y = 2$, but the shaded region is above this line. This is incorrect.

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Option (C): $x \geq 0$, $y \geq 0$, $2x + y \geq 2$, $x + 2y \leq 8$, $x - y \leq 1$

This describes the exact bounded region with all five constraints matching the shaded region perfectly. This is correct.

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Option (D): $x + 2y \geq 8$, $x - y \geq 1$

This is missing $x \geq 0$, $y \geq 0$ and has wrong inequality directions. This is incorrect.

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Option (C) gives the complete description of the exact bounded region, while Option (A) gives a partial description of a larger region that contains the shaded area. Both describe inequalities satisfied by the shaded region.

The answer is (A) and (C) only.