The feasible region determined by the constraints \(x + y \le 4\), \(x + 2y \ge 8\), \(x \ge 0\), \(y \ge 0\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → consists of exactly one point
Let (x, y) be a feasible point. Subtracting the first constraint from the second,
\(y = (x + 2y) - (x + y) \ge 8 - 4 = 4\)
But \(x + y \le 4\) with \(x \ge 0\) gives
\(y \le 4 - x \le 4\)
So \(y = 4\) and \(x = 0\). The point (0, 4) satisfies all four constraints, so the feasible region is the single point (0, 4), where the two lines \(x + y = 4\) and \(x + 2y = 8\) meet on the y-axis.
Why the other options are incorrect
Option (1) is empty — Incorrect. (0, 4) satisfies every constraint: \(4 \le 4\) and \(8 \ge 8\).
Option (2) is a line segment — Incorrect. Every feasible point must have \(x = 0\) and \(y = 4\), so there is only one point.
Option (4) is unbounded — Incorrect. \(x + y \le 4\) keeps every feasible point inside a triangle near the origin.