Region of Convergence
According to the properties of Z-transform, the ROC of sum of two signals is the intersection of their individual ROCs.

If you look at the signal 
$$ x[n] = (\frac{1}{3})^n u[n] + (2)^n u[-n-1] $$ 

you see that it consists of two signals; one right sided $(\frac{1}{3})^n u[n]$ with a pole at $z = 1/3$ and whose Z-transform is $X(z) = 1/(1 - (1/3)z^{-1})$ and with region of convergence ROC1: $ |z| &gt; 1/3$ and the other is a left sided one $(2)^n u[-n-1]$ with a pole at $z=2$ whose Z-transform is $-1/(1-2z^{-1})$ and with ROC2 as $ |z| &lt; 2$. Therefore, the intersection of their ROCs is
$$ROC_x = ROC1 \cap ROC2 = \{ 1/3 &lt; |z| \} \cap \{|z| &lt; 2 \} = \{ 1/3 &lt; |z| &lt; 2 \} $$  

similarly for the signal $y[n]$.