1. **Problem statement:** Find the coordinates of point $X$ on the line segment $\overline{RS}$ such that $X$ is $\frac{1}{6}$ of the distance from $R$ to $S$. 2. **Given:** - Coordinates of $R$ are $(-4, 2)$. - Coordinates of $S$ are $(2, -2)$. - Point $X$ divides $\overline{RS}$ at $\frac{1}{6}$ of the distance from $R$ to $S$. 3. **Formula used:** The coordinates of a point dividing a segment between points $R(x_1, y_1)$ and $S(x_2, y_2)$ in the ratio $m:n$ are given by: $$ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$ 4. **Applying the formula:** Since $X$ is $\frac{1}{6}$ of the distance from $R$ to $S$, the ratio is $1:5$ (from $R$ to $X$ is 1 part, from $X$ to $S$ is 5 parts). Coordinates of $X$: $$ \left( \frac{1 \times 2 + 5 \times (-4)}{1+5}, \frac{1 \times (-2) + 5 \times 2}{1+5} \right) = \left( \frac{2 - 20}{6}, \frac{-2 + 10}{6} \right) = \left( \frac{-18}{6}, \frac{8}{6} \right) $$ 5. **Simplify fractions:** $$ \left( \cancel{\frac{-18}{6}} \to -3, \cancel{\frac{8}{6}} \to \frac{4}{3} \right) $$ 6. **Final answer:** The coordinates of point $X$ are $\boxed{\left(-3, \frac{4}{3}\right)}$.