Question

Which of the following set of constraints represents the feasible region (shaded portion) in the figure given below?

• A. \(x+y \le 2,\; x+2y \le 3,\; x \ge 0,\; y \ge 0\)
• B. \(x+y \ge 2,\; x+2y \le 3,\; x \ge 0,\; y \ge 0\)
• C. \(x+y \le 2,\; x+2y \ge 3,\; x \ge 0,\; y \ge 0\)
• D. \(x+y \ge 2,\; x+2y \ge 3,\; x \ge 0,\; y \ge 0\)

Hint:

In feasible region problems, always test a simple point (like origin) to decide inequality direction.

Answer 1

The Correct Option is D

Concept: A feasible region is determined by identifying the correct side of each boundary line in a coordinate plane. Each line divides the plane into two half-planes, and the shaded region satisfies all inequalities simultaneously.

Step 1: {Identify boundary lines from the graph
From the given figure, the boundary lines are: \[ x+y=2 \quad \text{and} \quad x+2y=3 \] These lines pass through: \[ (2,0), (0,2) \quad \text{and} \quad (3,0), (0,\tfrac{3}{2}) \]

Step 2: {Determine shaded region
The shaded region lies in the first quadrant, hence: \[ x \ge 0,\quad y \ge 0 \] From the graph, the region is above both lines, so: \[ x+y \ge 2 \] \[ x+2y \ge 3 \]

Step 3: {Final constraint set
\[ x+y \ge 2,\quad x+2y \ge 3,\quad x \ge 0,\quad y \ge 0 \] Thus, it matches option (D).