How to justify a theorem logically?

Question

I had to justify the following theorem :

If $x + y \ge 100$ then justify that $x\ge 50$ or $y\ge 50$.

How can I do this ? I cant find any clue to solve this math . Please help me .

Answer 1

You can do it 'by contradiction'. If both $x<50$ and $y<50$, then $x+y < 50 + 50 \implies x+y < 100$, which contradicts our initial hypothesis of $x+y \geqslant 100$.

Answer 2

Assume the contrary, then x<50 and y<50, adding both inequalities, we get x+y<100, a contradiction. Hence proved.

Answer 3

Do a case analysis on either $x\geq y$ or $y > x$. In the $x \geq y$ case, $x+x \geq x+y$ therefore $2x\geq 100$ so $x \geq 50$ and thus, trivially, $x\geq 50\lor y\geq 50$. The $y > x$ case proceeds similarly.