The solution set of the inequation \( 2x+3y\gt 12 \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Open half plane not containing the origin
The line 2x + 3y = 12 divides the plane into two half planes. The inequality is strict, so points on the line are excluded (the half plane is open).
At the origin, 2(0) + 3(0) = 0, which is not greater than 12. So the solution set is the open half plane not containing the origin.
Why the other options are incorrect
Option (1) \( xy \)- plane except the points lying on the line \( 2x+3y=12 \) — Incorrect. This is the solution of \( 2x+3y\ne 12 \); it also contains points such as the origin, where \( 2x+3y\lt 12 \).
Option (2) Open half plane containing the origin — Incorrect. The origin does not satisfy 2x + 3y > 12, so its half plane is not the solution.
Option (4) \( xy \)-plane with all the points lying on the line \( 2x+3y=12 \) — Incorrect. The whole plane contains the origin and the line itself, and neither satisfies \( 2x+3y\gt 12 \).