To solve for \( x \) in the equation
\[ \frac{8}{11} \left( x + \frac{4}{5} \right) = \frac{24}{55}, \]
we start by eliminating the fraction on the left side. We can do this by multiplying both sides of the equation by \( 11 \):
\[ 8 \left( x + \frac{4}{5} \right) = \frac{24}{55} \cdot 11. \]
Calculating the right side:
\[ \frac{24 \cdot 11}{55} = \frac{264}{55} = \frac{264 \div 11}{55 \div 11} = \frac{24}{5}. \]
Now, our equation becomes:
\[ 8 \left( x + \frac{4}{5} \right) = \frac{24}{5}. \]
Next, we divide both sides by \( 8 \):
\[ x + \frac{4}{5} = \frac{24}{5} \cdot \frac{1}{8} = \frac{24}{40} = \frac{3}{5}. \]
Now, we isolate \( x \) by subtracting \( \frac{4}{5} \) from both sides:
\[ x = \frac{3}{5} - \frac{4}{5} = \frac{3 - 4}{5} = \frac{-1}{5}. \]
Thus, the solution for \( x \) is
\[ \boxed{-\frac{1}{5}}. \]