Question

find a counterexample of for all real numbers x,-x​

Answer 1

Hint: 2–√−2–√=02−2=0 is rational.

Your counterexample works fine. I think a better version would be

(2–√−1)+(1)=2–√

(2−1)+(1)=2

is clearly irrational, whereas 11 is rational. Otherwise you would have to justify why the sum of an irrational and a rational number is irrational.