The vertices of a triangle are $(1,3), (5,0)$ and $(-1,2)$. Which of the following inequalities will be satisfied by all points lying inside the triangle?
Step-by-Step Solution
Key Concept: A linear inequality holds for all points in a triangle if and only if all three vertices satisfy it with the same sign.
For $f(x,y) = 3x + 2y$, all three vertices satisfy $3x + 2y \geq 0$, so the inequality holds for all interior points. For $f(x,y) = 2x - 3y - 12$, all vertices satisfy $2x - 3y - 12 \leq 0$, so the inequality holds throughout. For $f(x,y) = x + y - 6$, vertices $(1,3)$ and $(-1,2)$ give opposite signs, so some interior points violate $2x - y \geq 0$.
Correct Answer: 1,2,3