1. The problem statement, all variables and given/known data Prove that for each real number x, (x+sqrt(2)) is irrational or (-x+sqrt(2)) is irrational. 2. Relevant equations We have already proven sqrt(2) is irrational and a rational+an irrational=irrational. 3. The attempt at a solution Proof by contradiction. Assume (x+sqrt(2)) or (-x+sqrt(2)) is rational. First set (x+sqrt(2))=(m/n) for some integers m and n. I get stuck here at where to go with the contradiction.