I know that for any two real numbers $a$ and $b$, that $$|a||b| = |ab|$$ And so I'd imagine that, assuming $x\in\mathbb{R}$, $$|x-a||x-b|=|(x-a)(x-b)|$$ since what is inside the absolute values are still real numbers.
Still, I have not been able to find out if this is true with Google.
As an example, is the following statement true?
$$\frac{|x^2-1|}{|x+1|} = \left|\frac{x^2-1}{x+1}\right| = |x-1|$$
The statement given to you, i.e. $\left|\frac{x^2-1}{x+1}\right| = |x-1|$ is true, except at the point $x = -1$ where the left hand side is not defined.
The point is, if $x$ is a variable taking values in some domain, then we can reference attributes/properties of $x$ seen in objects belonging to that domain (and that domain only). So $x-a$ and $x-b$ are real numbers even if they are not fixed real numbers, and hence the claim $|(x-a)(x-b)| = |x-a| |x-b|$ applies.
Note : This fact is also true for $x$ varying over the complex numbers, where the absolute value is the complex absolute value ($|x+iy| = \sqrt{x^2 + y^2}$), an extension of the real absolute value.
