To solve for \( x \) in the equation
\[ \frac{8}{11}(x + \frac{4}{5}) = \frac{24}{55}, \]
first, we can eliminate the fraction by multiplying both sides of the equation by \( 11 \):
\[ 8(x + \frac{4}{5}) = \frac{24}{55} \cdot 11. \]
Calculating the right side:
\[ \frac{24 \cdot 11}{55} = \frac{264}{55} = \frac{24 \cdot 11 / 5}{11 / 5} = \frac{264 / 5}{1} = \frac{264}{5}. \]
So we have
\[ 8(x + \frac{4}{5}) = \frac{264}{5}. \]
Now, divide both sides by \( 8 \):
\[ x + \frac{4}{5} = \frac{264}{5 \cdot 8} = \frac{264}{40} = \frac{33}{5}. \]
Next, to isolate \( x \), subtract \( \frac{4}{5} \) from both sides:
\[ x = \frac{33}{5} - \frac{4}{5} = \frac{33 - 4}{5} = \frac{29}{5}. \]
Thus, the solution is:
\[ x = \frac{29}{5}. \]
In decimal form, that is:
\[ x = 5.8. \]