Does $|x|=\pm x?$ And what is the graph of $y=|x|?$ And other questions.

Question

My use of "or" is as this: if $a$ or $b$ is true then $a$, and $b$ can be true together or either one of them is true and the other is false. I define $\pm x=+x$ or $-x.$

With these definitions, I want to ask

1. What is the graph of $y=\pm x?$
2. Does $|x|=\pm x,$ in general? In general, because it is true for values like $x=5$ i.e. $|5|=\pm 5$ is true.
3. If we replace $|x|\rightarrow \pm x$ in a sentence then will the new statement be equivalent to the original one? For e.g. is it true $y=|x|\iff y=\pm x?$
4. Is it true that $|x|=|y|\implies x=\pm y?$

Answer 1

($\bigstar$) Abosulute Value ($\bigstar$)
So when it comes to the graph you can see an image using Desmos here:

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Explanations

When it comes to $|x|=y$ it is always true that $\ x=\pm y $.
It is true that: |-5|=|5|=5 so |5|=$\pm 5$ is wrong.
it is also true that $y=|x|\implies y=\pm x$ so regarding to your question 4, replacing it's wrong.
Last but not least, it is true that $|x|=|y|\implies x=\pm y$ but it also true that:$|x|=|y|\implies y=\pm x$.