Pugh's exercise on Dedekind cuts addition

Question

I am trying to solve the following exercise:

Let $x=A|B$ and $x'=A'|B'$ be cuts in $\mathbb{Q}$. Show that although $B+B'$ is disjoint from $A+A'$, it may happen in degenerate cases that $\mathbb{Q}$ is not the union of $A+A'$ and $B+B'$.

The first assertion is straightforward, but I have not been able to handle the second one. It does not work for rational cuts or $\sqrt{2}+\sqrt{2}$.

Answer 1

HINT: Try $\sqrt2$ and $-\sqrt2$. More generally, try any two irrationals whose sum is rational.