jmc

We factor the denominator in the left-hand side to get $$\frac{5x-16}{(x-2)(x-5)}=\frac{A}{x-2}+\frac{B}{x-5}.$$We then multiply both sides by $(x-2)(x-5)$, to get $$5x-16=A(x-5)+B(x-2).$$We can solve for $A$ and $B$ by substituting suitable values of $x$. For example, setting $x=2$, the equation becomes $-6=-3A$, so $A=2$. Setting $x=5$, the equation becomes $9=3B$, so $B=3$. Therefore, $A+B=2+3=\boxed{5}$.