What I am asking is a more general question though: why does a contradiction arise when using the method?
It's quite obvious that squaring a number greater than 1 or less than its opposite -1 will be greater than 1. I understand that .
My question is: why do we even have one answer which is wrong (contradictory) from this method? Is it a problem of the method, a problem with the domain of the question, or not a problem at all, and just a contradiction to be rationally discarded because it's not useful?
Does the contradiction simply mean that the premises (x + 1) < 0 and (x - 1) > 0 were not correct, and the other premises (x + 1) > 0 and (x - 1) < 0 were?
I want to be generally aware of how I should approach math problems.