Let $f(x)$ and $g(x)$ be two functions.
Does the following property hold true? $$|f(x) - g(x)| = |g(x) - f(x)|$$
On the surface, it would seem like it, but then some other properties of absolute values don't seem to hold.
$$|f(x) - g(x)| \le a$$ $$-a \le f(x) - g(x) \le a$$
But this doesn't seem to be equivalent to the following $$|g(x) - f(x) | \le a$$ $$-a \le g(x) - f(x) \le a$$
Why doesn't the property $|f(x) - g(x)| = |g(x) - f(x)|$ hold true?
Yes. $$|a-b|=|-(b-a)|=|-1||b-a|=|b-a|,$$ for any $a,b\in V$, where $V$ is normed vector space, and $|\cdot|$ is norm (which is absolute value if $V=\mathbb{R}$) on $V$.