<s>[INST] Could someone elaborate on this algebraic transformation? I have a solution for an exercise and one part of it is not clear to me:
$
\frac{\frac{1}{\sqrt{x - 1}} - 1}{x - 2} = \frac{1 - \sqrt{x - 1}}{(x - 2)\sqrt{x - 1}}
$
Could anyone explain, please, how the result was obtained?
The full solution is this:
$
\frac{\frac{1}{\sqrt{x - 1}} - 1}{x - 2} = \frac{1 - \sqrt{x - 1}}{(x - 2)\sqrt{x - 1}} * \frac{1 + \sqrt{x - 1}}{1 + \sqrt{x - 1}} = \frac{2 - x}{(x - 2)\sqrt{x - 1}(1 + \sqrt{x - 1})} = \frac{-1}{\sqrt{x - 1}(1 + \sqrt{x - 1})}, x \ne 2
$
 [/INST] You simply multiply the top and bottom by $\sqrt{x-1}$:
$$\frac{\frac{1}{\sqrt{x-1}}-1}{x-2}=\frac{\Bigl(\frac{1}{\sqrt{x-1}}-1\Bigr)\sqrt{x-1}}{(x-2)\sqrt{x-1}}=\frac{1-\sqrt{x-1}}{(x-2)\sqrt{x-1}}$$
Edit: This is valid as long as the thing you are multiplying top and bottom by is not $0$. In this case, $\sqrt{x-1} \neq 0$ for or else the thing you were given containing a $\frac{1}{\sqrt{x-1}}$ would have been undefined in the first place.
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