How to solve $2x-2-(\ln(x-1))(x-1)>0$

Question

How to solve $2x-2-(\ln(x-1))(x-1)>0$.

Alpha says the solution is $1<x<1+e^2$

Answer 1

Guide:

Clearly, we need $x>1$ (justify why)

We have

$$2(x-1)-(\ln(x-1))(x-1)>0$$ $$(x-1)(2-\ln (x-1))>0$$

As mentioned earlier, we have $x-1>0$, so we can divide both side by $x-1$. Try to complete the task after that.