Consider the following statements about the graph of linear inequalities in x and y, where a and b are not both zero:
(A) The graph of \(ax + by \le c\) is the half-plane on one side of the line \(ax + by = c\), together with the points of the line.
(B) If \(c \gt 0\), the origin lies in the region \(ax + by \le c\).
(C) The region \(ax + by \ge c\) always contains the origin.
(D) The conditions \(x \ge 0, y \ge 0\) keep the feasible region in the first quadrant, including the parts of the axes that bound it.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
The line \(ax + by = c\) divides the plane into two half-planes. A point lies in a region exactly when its coordinates make the inequality true; at the origin the left side \(ax + by\) is 0.
So (A), (B) and (D) are true, and (C) is false.
Checking the statements
(A) The graph of \(ax + by \le c\) is the half-plane on one side of the line \(ax + by = c\), together with the points of the line. — Correct. The sign ≤ includes the line itself.
(B) If \(c \gt 0\), the origin lies in the region \(ax + by \le c\). — Correct. At (0, 0): \(0 \le c\) is true.
(C) The region \(ax + by \ge c\) always contains the origin. — Incorrect. For \(x + y \ge 2\): \(0 \ge 2\) is false at (0, 0).
(D) The conditions \(x \ge 0, y \ge 0\) keep the feasible region in the first quadrant, including the parts of the axes that bound it. — Correct. Both coordinates are zero or positive.