Some Results For Absolute Value

A handful of algebraic identities about absolute value are used throughout the rest of the chapter:

1. If $|y+x|=|x-y|$ for $x,y\in R$, then $xy=0$ (one of $x,y$ must be $0$).
2. $|xy|=|x|\,|y|$ for any $x,y\in R$ -- the absolute value of a product is the product of the absolute values.
3. $\left|\dfrac xy\right|=\dfrac{|x|}{|y|}$ for $x,y\in R,\ y\ne0$ -- likewise for quotients. …