Express this sum as a common fraction: $.\overline{8} + .\overline{2}$
In general, to express the number $0.\overline{n}$ as a fraction, we call it $x$ and subtract it from $10x$: $$\begin{array}{r r c r@{}l}
&10x &=& n&.nnnnn\ldots \\
- &x &=& 0&.nnnnn\ldots \\
\hline
&9x &=& n &
\end{array}$$ This shows that $0.\overline{n} = \frac{n}{9}$.

Hence, our original problem reduces to computing $\frac 89 + \frac 29 = \boxed{\frac{10}{9}}$.