jmc

Convert both decimals into fractions. $$\begin{array}{rl}x& =.\overline{2}\\ \Rightarrow 10x& =2.\overline{2}\\ \Rightarrow 9x& =2\\ \Rightarrow x& =\frac{2}{9}.\end{array}$$Likewise, $.\overline{6}=\frac{6}{9}$. Adding the two fractions yields $\frac{2}{9}+\frac{6}{9}=\frac{8}{9}$. The reciprocal of this is $\frac{1}{\frac{8}{9}}=\frac{9}{8}$. To convert this to a decimal, we need to multiply the numerator and denominator by 125. Doing so, we get $$\frac{9}{8}\cdot \frac{125}{125}=\frac{9\cdot 125}{8\cdot 125}=\frac{1125}{1000}=\boxed{1.125}.$$