Introductory abstract algebra, solve $|x − e| = x$ to find whether an identity element exists.

Question

I'm not sure how to solve the equation $$|x − e| = x,$$ where $e$ is the identity element. The book I'm reading gives the following proof of $$x * y = |x − y|$$ not having an identity element, but I don't get how they got to it, i.e. I don't get how to expand the absolute value in this case:

$$x ∗ e = x$$

$$|x − e| = x$$

$$e = 2x$$

Edit: The set is $\mathbb{R}$

Edit2: I'm looking for a step by step expansion of absolute value. I think in the negative case e should be 0, and in the positive it should be 2x, but I'm not sure how i would write it down in an answer.

Answer 1

It is not entirely clear what you are doing.

What is your set and what is your group operation?

It looks like you group operation $*$ is:

$x*y = |x-y|$

What is the identity element in the group.

You identified $e = 2x$ as a candidate. But this creates problems. If $2x$ is the identity and $y$ is also in the set then it must be the case that $y*2x = y$.

Futhermore, if $2x$ is the identity then $2x$ must be in the set and $2x*2x = 2x$.

$e = 0$ must be the identity in this problem.

Answer 2

What is was looking for is along those lines: $$x-e>0 \: => |x-e|=x-e \:=> x-e=x, e=0$$

$$x-e<0 \:=> |x-e|=-(x-e) \:=> -x+e=x, e=2x$$

$0 \neq 2x$, hence there is no identity element.

Is that valid?