jmc

The sum of an infinite geometric series with first term $a$ and common ratio $r$ is $\frac{a}{1-r}$. Thus the sum of the first series is

$$\frac{1}{1-\frac{1}{2}}$$And the sum of the second series is

$$\frac{1}{1+\frac{1}{2}}$$Multiplying these, we get

$$\frac{1}{1-{\left(\frac{1}{2}\right)}^2}=\frac{1}{1-\frac{1}{4}}$$So $x=\boxed{4}$.