2008 AMC 12A Problem 2

Using a common denominator, $$\dfrac{1}{2} + \dfrac{2}{3} = \dfrac{3}{6} + \dfrac{4}{6} = \dfrac{7}{6}.$$ The reciprocal of $\dfrac{7}{6}$ is $\dfrac{6}{7}.$ Thus, A is the correct answer.