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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?

rst_tk
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    Note that $\left( -a,a \right) = {x \in \Bbb R \mid a\leq x\leq b}$ is already wrong. – Martin R Jun 10 '20 at 11:30
  • Yes sorry, corrected it – rst_tk Jun 10 '20 at 11:32
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    Check https://math.stackexchange.com/q/2435975 or https://math.stackexchange.com/q/1035964 or https://math.stackexchange.com/q/2829096 or https://math.stackexchange.com/q/3201277 – Martin R Jun 10 '20 at 11:35
  • I wouldn't say they're equal. $|x|<a$ suggests there exists some specific $x\in\mathbb R$ whose absolute value is less than $a$. It doesn't specifically refer to an interval. And if you saw it outside of any other context, I think you's be more likely to interpret it as referring to some specific $x$ rather than an interval. – John Forkosh Jun 10 '20 at 11:36
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    @JohnForkosh More precise would be {$x\in \mathbb R\ |\ -a<x<a$} or {$x\in \mathbb R\ |\ |x|<a$} – Peter Jun 10 '20 at 11:38
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    @Peter Sure. But note that the op already said it exactly that way (after three tries:). His specific question's in the Subject. And I think the answer's just a matter of interpretation, like I suggested. So the op himself will have to clarify exactly what he's asking. – John Forkosh Jun 10 '20 at 11:42

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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 $a$.

Peter
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  • but how can $x$ get negative? I thought the abs of $x$ can only be positive.. so why is the lower bound of $-a$ relevant for the interval? – rst_tk Jun 10 '20 at 11:33
  • While the absolute value of $x$ cannot be negative, it is still possible for $x$ itself to be negative. As an example, if $|x|<10$, then it is possible for $x$ to be negative, e.g. $x$ could be $-9$, since $|-9| < 10$. – Minus One-Twelfth Jun 10 '20 at 11:34
  • It is true that $|x|$ cannot be negative, but $x$ can be. – Peter Jun 10 '20 at 11:36
  • i got it! thanks – rst_tk Jun 10 '20 at 11:38
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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!

Anas A. Ibrahim
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