Someone can explain this interval?

Question

I know that $ | a | < b$ it is a interval, where $ a $ and $ b $ is an generic expression, but how i can understand this? I want to transform this on the interval like this $ a < b < -a $ (or something like this) but i don't have idea how to make this.

Answer 1

The definition of $|a|$ is $$|a|=\begin{cases}a,& a\geq0\\-a,&a<0\end{cases}$$

Therefore $|a|<b$ means that either

$$0\leq a<b$$

or

$$a<0, -a<b$$

which is the same as $-b<a<0$.

Since there is an or all valid cases are the union of both cases.

The union of $-b<a<0$ and $0\leq a<b$ is $$-b<a<b$$

Answer 2

The inequality $$|a|<b$$ means that the absolute value of $a$ is strictly less than $b$; that is, $a$ is between $\pm b$.

Hence we have $$-b<a<b$$

Answer 3

Using the definition of absolute value the inequality $|a|<b$, is impossible for $b<0$ and, if $b\ge 0$ is equivalent to the couple of systems:

$$ \begin{cases} a\ge 0\\ a<b \end{cases} \quad \mbox{or}\quad \begin{cases} a< 0\\ -a<b \end{cases} $$

can you solve from here?