The solution set of the inequation \(4x + 3y \gt 5\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → open half plane not containing the origin
The line \(4x + 3y = 5\) divides the plane into two half planes, and the inequation holds on one of them. Testing the origin:
\(4(0) + 3(0) = 0\), and \(0 \gt 5\) is false.
So the solution set is the half plane on the side of the line away from the origin. Since the inequality is strict (>, not ≥), the points of the line itself are not included, so the half plane is open.
Why the other options are incorrect
Option (1) half plane that contains the origin — Incorrect. The origin does not satisfy \(4x + 3y \gt 5\).
Option (3) whole xy-plane except the points lying on the line \(4x + 3y = 5\) — Incorrect. That set is \(4x + 3y \ne 5\); it also contains the points with \(4x + 3y \lt 5\), such as the origin.
Option (4) lying only on first quadrant — Incorrect. The solution set also has points outside the first quadrant, for example (5, −1), where \(20 - 3 = 17 \gt 5\).