We are given $a<x<b$, where $a$ and $b$ are constants. This means that $x$ belongs to the interval from $x=a$ to $x= b$ excluding $a$ and $b$.
What is the interval that $|x|$ (the absolute value of $x$) belongs to?
This question continually bothered me when i was working in $ε-δ$ proofs. I've found an answer and I just want to make sure it's correct. We take 3 cases.
Case 1: $b> 0$ but $a <0$, case 2: both $a<0$ and $b<0$, case 3 : $a>0$ and $b>0$.
The answer in case one is $0≤|x|<\max(|b|,|a|)$. The answer in 2 would be $|b|<|x|<|a|$. And the answer in 3 is quite straightforward $a<|x|<b$. Is this correct ?
As far as I can see it is indeed correct. Usually a little sketch is very useful in this cases to see what's going on