The feasible region of inequality 2x+3ygt6 is

The correct answer is Option (4) → Half-plane that neither contains origin nor the points on the line \( 2x+3y=6 \)

The line \( 2x+3y=6 \) divides the plane into two half-planes, and the inequality holds on one of them. Testing the origin:

    \( \displaystyle 2(0)+3(0)=0 \)

and \( 0\gt6 \) is false, so the solution is the half-plane not containing the origin.

The inequality is strict (\( \gt \), not \( \ge \)), so the points on the line \( 2x+3y=6 \) are excluded as well.