TITLE: Inequality with (1-x) as denominator
QUESTION [9 upvotes]: How do I solve $\frac{1}{x-1}>0$ for $x$?
If I multiply both sides with $x-1$ then becomes $1\gt 0$. I know it's wrong. How do I solve it?

REPLY [2 votes]: The quotient of real numbers is positive $\iff$ numerator and denominator are of the same sign. Numerator is $1$, that's positive, so denominator must be positive, too: $$x-1>0$$ so the solution is $$x>1$$ Done.