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} $.