$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
The region of convergence (ROC) of $X(z)$ is the set of all values of $z$ for which $X(z)$ converges.
In this case, $x[n] = \left( {{1 \over 3}} \right)^n u[n] – \left( {{1 \over 2}} \right)^n u[ – n – 1]$.
The ROC of $x[n]$ is the set of all values of $z$ for which
$$\sum_{n=-\infty}^{\infty} \left| \left( {{1 \over 3}} \right)^n u[n] – \left( {{1 \over 2}} \right)^n u[ – n – 1] \right| z^{-n} < \infty$$
This can be rewritten as
$$\sum_{n=-\infty}^{\infty} \left| \left( {{1 \over 3}} \right)^n \right| z^{-n} + \sum_{n=-\infty}^{\infty} \left| \left( {{1 \over 2}} \right)^n \right| z^{-n} < \infty$$
The first sum converges if and only if $\left| z \right| > {1 \over 3}$. The second sum converges if and only if $\left| z \right| > {1 \over 2}$. Therefore, the ROC of $x[n]$ is the set of all values of $z$ for which $\left| z \right| > \max \left( {{1 \over 3}}, {{1 \over 2}} \right) = {1 \over 2}$. This is equivalent to $\left| z \right| > {1 \over 2}$.