The inequality 3x+5y<4 represents a region in the xy-plane. The related equation 3x+5y=4 is a straight line that acts as a boundary.
The inequality has a strict inequality sign "<" (not "≤"), which means the boundary line is not included in the solution set. This makes it an open region.
This eliminates the option of a closed half plane.
The inequality 3x+5y<4 divides the plane into two parts:
- Points where 3x+5y<4 (one half)
- Points where 3x+5y>4 (other half)
The solution is a half plane, not the whole plane.
This eliminates the option of the whole xy-plane.
To determine which half plane, test the origin (0,0) in the inequality:
3(0)+5(0)<4
0<4
This is true, so the origin satisfies the inequality and is in the solution set.
The solution set is an open half plane containing the origin.