A doubt about a delta estimation problem

A doubt about a delta estimation problem

I am able to solve these delta estimation problems, but I don't grasp the
idea behind them. Here goes the problem:
We have $f(x)=x^2$ and $a=5$
Determine $\delta > 0$ where $ 0 < \left| x -(-5) \right| < \delta $, then
$ \left| f(x)-25 \right| < \frac{1}{20} $
I know the process:
$$ \left| f(x)-25 \right| < \frac{1}{20} \\ \left| x^2-25 \right| <
\frac{1}{20} \\ \left| x-5 \right|\left| x+5 \right| < \frac{1}{20} \\ $$
$$ \text{Then, we assume that } \delta_o=1 \\ \left| x+5 \right| <
\delta_o=1 \\ -\delta_o < x+5 < \delta_o \\ -1 < x+5 < 1 \\ -11 < x-5 < -9
\\ \text{Taking us to } \left| x-5 \right|<11 \\ $$ $$ \left| x-5 \right|
\left| x+5 \right| \leq 11\left| x+5 \right| < \frac{1}{20} \\ \left| x+5
\right| < \frac{1}{220} \\ \text{Finally we get that $\delta$ needs to be
the smaller number in the interval } (1, \frac{1}{220}) \text{ and $\delta
= \frac{1}{220}$} $$
I get lost, in the interpretation, when we assume that delta is $1$ and
its consequences. Why do we need to convert the expression in that block
to the one we want to eliminate from the equation that needs to be
satisfied? Why do we choose $11$, too? I was not shown the graphical
representation for this problem, by the way.