# Absolute Value Proofs: Products and Quotients

The absolute value of a number is its distance from 0. Thus, all absolute values are either positive or 0. That is, if we have a number $latex x$, then the absolute value of $latex x$ which can be written as $latex |x|$ is equal to

(a) $latex x$ if $latex x$ is positive

(b) $latex -x$ if $latex x$ is negative

(c) $latex 0$ if $latex x$ is 0.

Also, for any real number, $latex x^2$ is positive or 0 (if $latex x = 0$).  Therefore, $latex \sqrt{x^2}$ is

(a) $latex x$ if $latex x$ is positive

(b) $latex -x$ if $latex x$ is negative

(c) $latex 0$ if $latex x$ is 0.

If the $latex -x$ part is a bit confusing, consider take for example  Continue reading