Find a $\delta >0$ such that if $x< \delta $, then $|\frac{1}{x+1}-1|<0.05$. Could someone explain something to me about the following question?

Find a $\delta >0$ such that if $|x|< \delta $, then $|\frac{1}{x+1}-1|<0.05$.

I got the inequality $-0.0476\leq x\leq 0.0526$, and the textbook chose $\delta = 0.0476$. Could someone please explain why they chose this $\delta$? Why not $\delta = -0.0476$? Would it be too big since then it would be like $0.04997<0.05$?
 A: Note that to fulfill the condition obtained
$$-0.0476\leq x\leq 0.0526$$
we need to choose 
$$\delta=\min\{0.0476,0.0526\}>0$$
such that for 
$$|x|<\delta\implies -0.0476\leq x\leq 0.0526$$
indeed recall that
$$|x|<\delta\iff -\delta <x<\delta$$
A: You just have to find a $\delta$,
not the best one.
At least that is what you wrote.
$|\dfrac{1}{x+1}-1|
=|\dfrac{1-(x+1)}{x+1}|
=|\dfrac{x}{x+1}|
$,
so if
$|x| < \frac12$,
$|\dfrac{1}{x+1}-1|
\lt \dfrac{|x|}{\frac12}
=2|x|
$.
Therefore,
to make
$|\dfrac{1}{x+1}-1|
\lt .05$,
it is sufficient to make
$2|x| < .05$
or
$|x| < .025$.
This is not the best,
but it will do.
If the problem wants
the largest bound,
it should say so.
A: $$|x|\le 0.0476 \implies -0.0476\leq x\leq 0.0476 \implies-0.0476\leq x\leq 0.0526 $$
A: 
Why not δ=−0.0476? 

Because that is a negative number.
And $|\frac 1{x+1} -1| < -0.0476 < 0$ will never happen.
And the didn't chose $\delta = 0.0526$ because that is too big/not strict enough.
The number $x = -0.05$ is such that $|x|< 0.0526$ but
$|\frac 1{x+1} - 1| = |\frac 1{-.05+1} -1| = |\frac 1{.95} - 1| = |1.0526315789473684210526315789474 - 1| = .0526315789473684210526315789474 > .05$.
If we have $-0.0476 < x < 0.0526$ then we have CAN'T have $x < -0.0476$, we must choose $\delta$ to be the MINIMUM of $|-0.0476|$ and $|0.0526|$.
Now you may be saying: but then we are losing all the $0.476 < x < 0.0526$ for which $x$ would be okay.  But that's okay, we don't need find all possible $x$.  Just a range of $x$ where $|\frac 1{x+1} - 1|$ must be less than $.05$.  Not could be (as $0.476 < x < 0.0526$ could be) but must be (as $-0.0526< x < -0.0476$) can't be.
