Which one of the following represent the correct feasible region (shaded region) determined by the following constraints of an LPP?
\( x+y\ge 4,3x+3y\ge 18,x\ge 0,y\ge 0 \) where \( z=2x+3y \) be the objective function.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4)
The second constraint simplifies:
\( \displaystyle 3x+3y\ge 18 \)
\( \displaystyle x+y\ge 6 \)
Every point with \( x+y\ge 6 \) automatically has \( x+y\ge 4 \), so the first constraint adds nothing. The feasible region is the part of the first quadrant (\( x\ge 0,\ y\ge 0 \)) on or above the line x + y = 6, on the side away from the origin (the origin gives 0 ≥ 6, false). It is unbounded.
Only the graph in Option (4) shades the region beyond the outer line x + y = 6 and nothing else. The objective function z does not affect the feasible region.
Why the other options are incorrect
Option (1) — Incorrect. Besides the correct region, it also shades the triangle \( x+y\le 4 \), where \( x+y\ge 6 \) fails.
Option (2) — Incorrect. It shades the strip between the lines, \( 4\le x+y\le 6 \), where \( x+y\ge 6 \) fails (except on the line itself).
Option (3) — Incorrect. It shades \( x+y\le 4 \), the side of the inner line towards the origin, which violates both constraints.