In each option below, the constraints \(x \ge 0\) and \(y \ge 0\) also hold. For which set of constraints is the feasible region unbounded?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(x + 2y \ge 6\), \(x - y \le 2\)
A feasible region is bounded if it can be enclosed in a circle, that is, x and y cannot become as large as we like. In option (2) take x = 0 and any large y:
\(x + 2y = 2y \ge 6\) when \(y \ge 3\)
\(x - y = -y \le 2\)
Every point (0, y) with \(y \ge 3\) is feasible, so the region goes on without end upwards: it is unbounded.
Why the other options are incorrect
Option (1) \(x + y \le 8\), \(2x + y \le 10\) — Incorrect. \(x + y \le 8\) with \(x, y \ge 0\) keeps x and y at most 8, so the region is bounded.
Option (3) \(x + y \ge 4\), \(x \le 6\), \(y \le 5\) — Incorrect. \(x \le 6\) and \(y \le 5\) keep the region inside a rectangle, so it is bounded.
Option (4) \(x + y \le 5\), \(x \ge 1\) — Incorrect. \(x + y \le 5\) keeps x and y at most 5, so the region is bounded.