simplify the expression ${\sqrt{x - 2\sqrt{x -1}} +\sqrt{x + 2\sqrt{x -1}}}$ 
Simplify the expression ${\sqrt{x - 2\sqrt{x -1}} +\sqrt{x + 2\sqrt{x -1}}}$

The problem is from a not so well renowned book for calculus in India - Concepts of Functions & Calculus - Vikas Rahi, ISBN 9780070080805. The answer for this question is given as
$$\lvert \sqrt{x-1} -1  \rvert + \lvert \sqrt{x-1}+1 \rvert$$
My efforts weren't great at all, I tried rationalizing them but the denominator becomes very similar to the question, which is totally not helpful for simplification.

$$\begin{align}{\sqrt{x - 2\sqrt{x -1}} +\sqrt{x + 2\sqrt{x -1}}}&=T(x)>0&\end{align}$$
$$T^2(x)=2x+2 \sqrt{x^2-4x+4}$$
$$T^2(x)=2x+2|x-2|$$
$$\begin{align}\color {gold}{\boxed {\color{black}{T(x)=\sqrt{2x+2|x-2|}}}}\end{align}$$


*

*If $x≥2$, then $\begin{align}T(x)&=\sqrt{2x+2x-4}\\
&=2\sqrt{x-1}\end{align}$


*If $x<2$, then $\begin{align}T(x)=\sqrt{2x+4-2x}=2.\end{align}$
