The absolute value function: \[|x|=\begin{cases}x&x\geq0\\-x&x<0\end{cases}\]
Absolute value properties:
- \(|a|=|-a|\)
- \(|ab|=|a||b|\)
- \(|a\pm b|\leq|a|+|b|\) (The triangle inequality)
Consider \(|ab|=|a||b|\). There are four possible states: \((a,b<0);(a<0,b>0);(a>0,b<0);(a,b>0)\).
If \(a,b<0\): \(ab>0\implies|ab|=ab\). Also \(|a|=-a\) and \(|b|=-b\) thus \(|a||b|=(-a)(-b)=ab\).
If \(a<0,b>0\): \(ab<0\implies|ab|=-ab\). Then \(|a|=-a\) and \(|b|=b\) thus \(|a||b|=-ab\).
If \(a>0,b<0\): \(ab<0\implies|ab|=-ab\). Then \(|a|=a\) and \(|b|=-b\) thus \(|a||b|=-ab\).
If \(a,b>0\): \(ab>0\implies|ab|=ab\). Also \(|a|=a\) and \(|b|=b\) thus \(|a||b|=ab\).
QED