Example 11. When $x \geqslant 1, x \neq 2$, simplify
$$
y=(x+2 \sqrt{x-1})^{-\frac{1}{2}}+(x-2 \sqrt{x-1})^{-\frac{1}{2}}
$$
Solve $\because y^{2}=(x+2 \sqrt{x-1})-1$
$$
\begin{array}{l}
+2(x+2 \sqrt{x-1})^{-\frac{1}{2}}(x-2 \sqrt{x-1})-\frac{1}{2} \\
+(x-2 \sqrt{x-1})^{-1} \\
=\frac{2}{x^{2}-4 x+4}+\frac{2}{|x-2|} .
\end{array}
$$
(1) When $1 \leqslant x < 2$, hence $\mathrm{y}=\sqrt{\frac{4}{(\mathrm{x}-2)^{2}}}=\frac{2}{-(\mathrm{x}-2)}$
$$
=\frac{2}{2-x} \text {. }
$$
(2) When $x>2$, $y^{2}=\frac{2 x}{(x-2)^{2}}+\frac{2}{x-2}$
$$
=\frac{4(x-1)}{(x-2)^{2}} \text {. }
$$

Hence $y=\sqrt{\frac{4(x-1)}{(x-2)^{2}}}=\frac{2 \sqrt{x-1}}{x-2}$.