To solve the equation
\[ \frac{8}{11} \left( x + \frac{4}{5} \right) = \frac{24}{25}, \]
we start by isolating the term \(x\). First, we can eliminate the fraction on the left-hand side by multiplying both sides by \(\frac{11}{8}\):
\[ x + \frac{4}{5} = \frac{24}{25} \cdot \frac{11}{8}. \]
Calculating the right side:
\[ \frac{24 \cdot 11}{25 \cdot 8} = \frac{264}{200} = \frac{66}{50} = \frac{33}{25}. \]
So, we have:
\[ x + \frac{4}{5} = \frac{33}{25}. \]
Next, we need to subtract \(\frac{4}{5}\) from both sides. To do this, we convert \(\frac{4}{5}\) to have a common denominator of 25:
\[ \frac{4}{5} = \frac{20}{25}. \]
Now, substitute this back into the equation:
\[ x + \frac{20}{25} = \frac{33}{25}. \]
Subtract \(\frac{20}{25}\) from both sides:
\[ x = \frac{33}{25} - \frac{20}{25} = \frac{33 - 20}{25} = \frac{13}{25}. \]
Thus, the solution is:
\[ \boxed{\frac{13}{25}}. \]
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