How to write the proof for $|x|>a \implies x >a$ or $x<-a$

Question

Can you please help me with the proof of this? I kwon this is for the properties of the absolute value, $|\cdot|$, but i can't figurate how get the "or".

$|x|>a \implies x>a \; \;\text{or} \;\; x<-a$

How you get that conclusion and not

$|x|>a \implies x>a \; \;\text{and} \;\; x<-a$

What is the best form to write the proof ? and get the "or".

Answer 1

The definition of the absolute value is $$|x|:=\begin{cases}x&x\ge 0\\ or\\ -x&x<0\end{cases}.$$

Therefore $$|x|>a\implies \begin{cases}x>a&x\ge 0\\ or\\ -x>a&x<0\end{cases}\implies x>a\text{ or }x<-a.$$