To solve the equation
\[ \frac{8}{11} \left(x + \frac{4}{5}\right) = \frac{24}{55}, \]
we can start by eliminating the fraction on the left side. To do that, multiply both sides of the equation by \( \frac{11}{8} \):
\[ x + \frac{4}{5} = \frac{24}{55} \times \frac{11}{8}. \]
Let's calculate the right side:
\[ \frac{24 \times 11}{55 \times 8} = \frac{264}{440}. \]
Now, we simplify \( \frac{264}{440} \). Both numbers can be divided by 88:
\[ \frac{264 \div 88}{440 \div 88} = \frac{3}{5}. \]
So we have:
\[ x + \frac{4}{5} = \frac{3}{5}. \]
Next, we need to isolate \( x \) by subtracting \( \frac{4}{5} \) from both sides:
\[ x = \frac{3}{5} - \frac{4}{5} = \frac{3 - 4}{5} = -\frac{1}{5}. \]
Therefore, the solution for \( x \) is
\[ \boxed{-\frac{1}{5}}. \]