I have to determine whether or not the Reals has an identity element with respect to *, given that x * y = |x - y|.

To do this, we were taught to solve x * e = x for e.

So x * e = |x - e| = x, which implies that x - e = x OR x - e = -x.

Thus e = 0 OR e = 2x.

Now we want to check that e * x = x * e = x.

e=0:

e*x = 0*x = |0-x| = |-x| = x

x*e = x*0 = |x-0| = x

e=2x:

e*x = |2x-x| = x

x*e = |x -2x| = |-x| = x.

They both work, but it doesn't make sense that there are two identity elements. What am I missing? Thanks for any help...