inequality of variables without knowing their sign

Question

I would like to manipulate an inequality when the said inequality is in terms of variables and I do not know the signs of the variables.

For example, $x_1<x_2$ and I would like to divide both sides by $x_1$.

If $x_1<0 $, $x_1<x_2 \implies 1 >\frac{x_2}{x_1} $.

If $x_1>0 $, $x_1<x_2 \implies 1 < \frac{x_2}{x_1} $.

But, without haivng this knowledge, do I always have to split the inequality into two separate cases, as I attempt to simplify my expression?

Answer 1

Yes, you are correct. You need to consider both cases.