Question

The shaded region in the provided graph represents the solution set for which of the following systems of linear inequalities?

• A. \(2x+y\geq 2, x-y\geq 1, x+2y\leq 8, x\geq 0, y\geq 0\)
• B. \(x+2y\geq 2, x-y\geq 1, x+2y\leq 8, x\geq 0, y\geq 0\)
• C. \(2x+y\geq 2, x-y\leq 1, x+2y\leq 8, x\geq 0, y\geq 0\)
• D. \(2x+y\geq 2, x-y\leq 1, 2x+y\leq 8, x\geq 0, y\geq 0\)

Hint:

Find each boundary line from the labelled points and test a point in the shaded region.

Answer 1

The Correct Option is C

Step 1: Corner Points Check:
The corners of the shaded region are $E(0,2)$, $A(1,0)$, $B\left(\frac{10}3,\frac73\right)$ and $D(0,4)$.

Step 2: Test Each Constraint at Corners:
Option (C) constraints: $2x+y\geq2$ holds with equality at $E$ and $A$; $x-y\leq1$ holds with equality at $A$ and $B$; $x+2y\leq8$ holds with equality at $B$ and $D$. All four corners satisfy all three.

Step 3: Rule Out the Others:
At $E(0,2)$: option (A) requires $x-y=-2\geq1$, which is false. Option (B) has $x+2y\geq2$, true at E, but also needs $x-y\geq1$, false at E. Option (D) requires $2x+y\leq8$ and $x-y\leq1$, which hold at E, but then the upper edge would be $2x+y=8$, not through $D(0,4)$ and $B$. So (C).

Final Answer:
Option (C). \[ \boxed{\text{(C)}} \]