This is a homework assignment, please tell me if my proof is correct!
Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x + y = z.
Assume a and b are integers with GCD (a,b) =1, and that c is an irrational number. There exists a number d = (a/b) + c.
Assume that d is rational. Then d - (a/b) would be rational, what is a contradiction because c is irrational. Therefore, d is irrational.
To prove uniqueness we can use the fact that the addition of any two real numbers has only one result, then d is unique. Q.E.D.