Prove the statement: If x is irrational and y is irrational, the x + y is irrational.

Contrapositive: If x + y is rational, then x is rational or y is rational.

My best guess at an approach is to prove the contrapositive. This is how I would start:

Proof:

Suppose x + y is rational. Then, by defn., x + y = p/q for some intergers p,q where q ≠ 0...

I'm not sure what to do from here. How do I deduce that x is rational or y is rational?