Let $X,Y$ be rea random variables with cdf $F_X, F_Y$, respectively. I have to find an upper bound to $$\mathbb P(X+Y>t)\qquad t\in \mathbb R$$ without supposing the independence of $X,Y$, otherwise the solution is simply found by convolution.
My Idea is to define $$Z = t-X$$ so that the problem should reduce to the comparison between $$F_Z(x) = F_{t-X}(x)= 1-F_X(t-x);\qquad F_Y(x)$$ but then I cannot proceed in a riguros way.