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We present a polynomial-time approximation scheme (PTAS) for all Boolean arity-two maximum constraint satisfaction problems (Boolean Max-2-CSP) where the constraint graph is either interval or has bounded rank-width. This class of problems includes, for example, Max-Cut, Max-DiCut, Max-2-Lin(2), and Max-2-SAT. In particular, this settles the approximability of Max-Cut on interval graphs. To achieve these results, we show the following. 1. For any $\varepsilon > 0$, there is an $n^{\mathcal O(r \varepsilon^{-2})}$-time $(1 - \varepsilon$)-approximation for Boolean Max-2-CSP on graphs of rank-width at most $r$. 2. There is a polynomial-time algorithm that, for any interval graph $G = (V, E)$ and $\varepsilon > 0$, finds a subgraph $H = (V, F)$ of $G$ such that $|F| \geq (1 - \varepsilon) |E|$ and $H$ has rank-width at most $\operatorname{poly}(\varepsilon^{-1})$. Our techniques additionally extend to all bounded-alphabet arity-two max CSPs.