Solutions for inequality $\frac{1}{x} + \frac{1}{1-x} \gt 0$ How to find all real solutions for $$\frac{1}{x} + \frac{1}{1-x} \gt 0$$
I came up with $\frac{1}{x(1-x)} \gt 0$ implies $x(1-x)>0$ and finally ended with $0<x<1$ but the answer provided was $0<x<1$ or $x>1$. I tried sample values for $x>1$ but they don't satisfy the inequality.

The function $$f(x)=\frac{1}{x} + \frac{1}{1-x}$$ may only change where it's not defined, i.e., for $x\in\{0,1\}$ or in its zeroes, which don't exist.  Now check the corresponding three intervals $(-\infty,0)$, $(0,1)$ and $(1,\infty)$ for positivity.
