Solve the inequality $-1<\frac{x}{\log(x-1)(x-1)}<1$

Question

Can you help me solve the inequality:

$$-1<\dfrac{x}{\log(x-1)(x-1)}<1$$

I try:

$-1<\dfrac{x}{\log(x-1)(x-1)}$

$=\dfrac{-\log(x-1)(x-1)}{x}<0$

$=0<\log(x-1)$

$= \log(x-1)>0$

$= x-1>10^0$

$= x>2$

In the second inequation I'm having more problems

Answer 1

This inequality cannot be solved algebraically. It can be solved approximately using a graphing utility.

The function $f(x)=\dfrac{x}{\log\left((x-1)^2\right)}$ has a vertical asymptote at $x=2$ and has removeable discontinuities at $x=0$ and $x=1$ since $\lim_{x\to0}f(x)=-\frac{1}{2}\ln(10)$ and $\lim_{x\to1}f(x)=0$.

Although one may analyze the inequality further, ultimately one must resort to using non-elementary methods or graphing to find the solution.

The approximate solution of the inequality is $(0.251,1)\cup(1,1.24)$.

Here is a wide view and a detailed view of the function $f(x)=\dfrac{x}{\log\left((x-1)^2\right)}$ together with the graphs of $y=1$ and $y=-1$ with the intersection points indicated by their coordinates.