Why is the following simplification possible?

Question

I have seen the following simplification:

$$\left|\frac{1}{(-1-\frac{1}{n})^4 - 1}\right| = \frac{1}{\left|-1-\frac{1}{n}\right|^4 - 1}$$

I really don't have a clue why this is possible...

I am sorry but I can't give any ideas because I don't have any.

Thank you for advice.

FunkyPeanut

Answer 1

Note that $x \geq 0$ if and only if $|x|=x$. So we have $\frac{1}{(-1-\frac{1}{n})^4 - 1}\geq 0$.

Inequlity $(-1-\frac{1}{n})^4 - 1\geq 0$ implies that $1+\frac{1}{n}\geq 1$. Hence for every natural number $n$, we have $|\frac{1}{(-1-\frac{1}{n})^4 - 1}| = \frac{1}{|-1-\frac{1}{n}|^4 - 1}$.

Answer 2

Have you tried $\left|f(x)\right|=\sqrt{f(x)^2}$?