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: Boundary Lines From the Figure:
Line through \(E(0,2)\) and \(A(1,0)\): \(2x+y=2\).
Line through \(A(1,0)\) and \(B\left(\frac{10}3,\frac73\right)\): slope \(1\), so \(x-y=1\).
Line through \(D(0,4)\) and \(B\left(\frac{10}3,\frac73\right)\): \(x+2y=8\), since \(\frac{10}3+\frac{14}3=8\).

Step 2: Pick a Test Point:
The shaded region lies inside the polygon \(E A B D\), for example near \((1,2)\).
\(2x+y=4\geq2\), so \(2x+y\geq2\) holds.
\(x-y=-1\leq1\), so \(x-y\leq1\) holds.
\(x+2y=5\leq8\), so \(x+2y\leq8\) holds.

Step 3: Axis Conditions:
The region is in the first quadrant, so \(x\geq0,\ y\geq0\).

Step 4: Match:
The system is \(2x+y\geq2,\ x-y\leq1,\ x+2y\leq8,\ x\geq0,\ y\geq0\), which is option (C). Option (A) has \(x-y\geq1\), which would pick the region below the line \(AB\). Option (B) uses \(x+2y\geq2\), a line that is not in the figure. Option (D) uses \(2x+y\leq8\), also not in the figure.

Final Answer:
The system is option (C). \[ \boxed{\text{(C)}} \]