Why is $\left( -a,a \right)$ equal to $|x| < a$

Question

I have a problem understanding the equality of an open Interval as given $\left( -a,a \right) = \{x \in R | -a< x< a\}$ to say $|x| < a$..

Maybe someone can get me intuitive to understand that?

Answer 1

Since the end points are not contained in the interval, we have $|x|<a$ instead of $|x|\le a$. A number $x$ is between $-a$ and $a$ ($-a<x<a$) if and only if its absolute value is smaller than $a$.

Answer 2

I was going to explain it mathematically but since you asked for an intuitive understanding, I'll go for it:

Look at the graph of $|x|$ (tip: try to graph on Desmos), it looks like never-ending $2$-sides of a right-angled triangle at the origin and pointing upwards, with the $y$-axis as the bisector of this triangle.

Now if you put $y<a$ for some $a$, you'll notice that this restricts the interval of $x$ to $(-a,a)$ on the graph of $|x|$, so this settles it!