Which of the following sets of constraints for a linear programming p…

The correct answer is Option (3) → \(x+y \le 10,\ 2x+y \ge 40,\ x \ge 0,\ y \ge 0\)

A set of constraints has no feasible region when no point satisfies all of them together.

In Option (3), for any point with \(x \ge 0,\ y \ge 0\) and \(x + y \le 10\):

    \(2x + y = x + (x + y) \le 10 + 10 = 20\)

So 2x + y can never reach 40, and the constraint \(2x + y \ge 40\) cannot hold. The feasible region is empty.

The other sets each have at least one feasible point:

    \(\text{Option (1): } (0, 0) \text{ satisfies all four}\)
    \(\text{Option (2): } (30, 0) \text{ gives } 30 \le 50,\ 90 \ge 80\)
    \(\text{Option (4): } (0, 0) \text{ satisfies all four}\)