Consider the constraints \(2x + y \ge 4\), \(x + y \le 6\), \(x - y \le 2\), \(x \ge 0\), \(y \ge 0\), and the following statements:
(A) The origin satisfies \(x + y \le 6\) and \(x - y \le 2\), but not \(2x + y \ge 4\).
(B) The point (1, 1) lies in the feasible region.
(C) The point (3, 1) lies on the boundary line of \(x - y \le 2\) and is in the feasible region.
(D) The feasible region has exactly 3 corner points.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
The boundary lines meet the axes and each other at these points:
\(2x + y = 4\) and \(x - y = 2\) meet at (2, 0)
\(x + y = 6\) and \(x - y = 2\) meet at (4, 2)
\(x + y = 6\) meets the y-axis at (0, 6)
\(2x + y = 4\) meets the y-axis at (0, 4)
So the feasible region has the 4 corner points (2, 0), (4, 2), (0, 6) and (0, 4).
Checking the statements
(A) The origin satisfies \(x + y \le 6\) and \(x - y \le 2\), but not \(2x + y \ge 4\). — Correct. At (0, 0): \(0 \le 6\), \(0 \le 2\), but \(0 \ge 4\) is false.
(B) The point (1, 1) lies in the feasible region. — Incorrect. \(2(1) + 1 = 3 \lt 4\).
(C) The point (3, 1) lies on the boundary line of \(x - y \le 2\) and is in the feasible region. — Correct. \(3 - 1 = 2\), \(2(3) + 1 = 7 \ge 4\), \(3 + 1 = 4 \le 6\).
(D) The feasible region has exactly 3 corner points. — Incorrect. It has 4: (2, 0), (4, 2), (0, 6), (0, 4).