Question

Hello there. I just had one question dealing with binary operationsin abstract algebra. I need to see if this operation has anidentity element for all x in R. The operation is as follows:x*y=|x-y| This is what I did. I basically broke it up since thereis an absolute value. x*e=|x-e| = x-e if x> 0 or -(x-e) if xe,then x*e= x-e. We need to find e for x*e=x. So x-e=x and e=0. Nowmy first question is this: 1) If x>0, then why for thisequation, x-e=x, can x be values that are less than 0. It seemslike a contradiction to me. Is e=0 only valid for x>0? Thenafter that I tried out x*0= |x|= x if x>0 and -x if x<0.However, is it just a coincidence that it is x if x>0 and alsowe chose x*0 to be x-0 for x>0? I mean, it seems like either way( whether you use the |x| way or the (x-e) way, you get x as ananswer). My next question is that if we choose x<0 for the abovex*0=..., then doesn't e change? I am confused about this. I mean inone case we have |x|= -x if x<0, but if x<0, then e changes,doesn't it? Please just help me answer these questions and guide mestep by step through this problem. Thank you.