Absolute Value Algebra with inverses

Question

I noticed the following equality in some material regarding limits and infinite series.

$$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$

And I'm honestly stumped (and slightly ashamed) on how to algebraically go from the lefthand side to the righthand side. Any pointers?

Thanks!

Answer 1

It's just a little bit of algebra to get there.

\begin{align*} \left|\frac{x}{x+1} -1\right| &= \left|\frac{x}{x+1} - \frac{x+1}{x+1}\right| \\ &= \left|\frac{x-x-1}{x+1}\right| \\ &= \left|\frac{-1}{x+1}\right| \end{align*}

I hope that helps.

Answer 2

$$\frac{-1}{x+1}+1 = \frac{x}{x+1}$$