Prove an inequality $\ln(1-1/x)<2/(1-2x)$

Question

I need some help to prove this inequality:

$$\ln(1-1/x) < \frac{2}{1-2x}$$ with $$x > 1$$

I did plot the curve of $\ln(1-1/x)-2/(1-2x)$ and it's always in minus.

Many thanks in advance!

Answer 1

Let $$f(x)=\ln(1-1/x)-\frac{2}{1-2x} \implies f'(x)=\frac{1}{x(x-1)(1-2x)^2}>0, if ~ x>1.$$ So $f(x)$ is an increasing function for $x>1 \implies f(x)<f(\infty) \implies f(x)<0$, and hence the result.