Relation between conditional expectation and CDF

Question

Let X be a nonnegative random variable. We know:

1. $F_X(1)=0.7, F_X(3)=0.9$
2. $E(X-1)_+=5.5, E(X-3)_+=5$

Find $E(X|X \in (1,3])$.

I know that $EX = \int_0^\infty (1-F_X(x))dx$ and I have to use that, but I don't have a clue how. It would be nice when $E(X|X \in (1,3])=E(X-1)_+-E(X-3)_+$ but it isn't true.