Which ONE of the following options represents correctly ALL the constraints resulting in the feasible region for the given linear programming problem as shown in the figure?
Note: Figure not to scale
Show Hint
Find the line through each pair of boundary points, then check which side of each line the shaded triangle lies on.
Step 1: Pick a point clearly inside the shaded triangle.
Looking at the figure, a point like $(1, 1)$ sits well inside the shaded region, away from all three edges.
We can plug this point into each option and throw out any option that this point fails to satisfy.
Step 2: Test the top-boundary type of constraint.
$x_2 = 1$ at our test point, and the flat top of the triangle is at height 3, so a correct top constraint must allow $x_2 = 1$, meaning it should read $x_2 \le 3$, not $x_2 \ge 3$.
This already removes option (A), since it uses $x_2 \ge 3$, which our test point fails.
Step 3: Test the diagonal constraints with the same point.
For option (D), check $x_1 + x_2 \le 1$: at $(1,1)$ this gives $1 + 1 = 2$, which is not $\le 1$, so option (D) fails this test too.
For options (B) and (C), check $x_1 - x_2 \le 1$: at $(1,1)$ this gives $1 - 1 = 0 \le 1$, which passes for both.
Step 4: Use the top-edge height to separate (B) and (C).
The only difference left between (B) and (C) is $x_2 \le 3$ versus $x_2 \le 4$.
Since the figure clearly marks the flat top of the shaded region at height 3, right where the point $(4,3)$ sits, the correct bound must be $x_2 \le 3$, not $x_2 \le 4$.
Step 5: Confirm the remaining option.
This leaves option (B) as the only one consistent with both the test point and the marked height of the top edge in the figure.
$x_1 \ge 0, x_2 \ge 0, x_2 \le 3, x_1 - x_2 \le 1, x_1 + 0.5x_2 \ge 1$
Final Answer:
Testing a point inside the region and matching the marked top edge both confirm option (B).
\[ \boxed{\text{Option (B)}} \]
