$|2- (\sqrt{n^2+4n} - n)| ≥ \frac{1}{10}$

Question

Any suggestions how to solve the following equation:

$|2- \sqrt{n^2+4n} + n| ≥ \frac{1}{10}$

Thank you in advance.

Answer 1

Hint: observe that $$ \left|2-\sqrt{n^2+4n}+n\right|\ge\frac1{10}|\Longleftrightarrow\\ \left(2-\sqrt{n^2+4n}+n\ge\frac1{10}\right)\vee\left(2-\sqrt{n^2+4n}+n\le-\frac1{10}\right) $$