To solve the equation \( \frac{8}{11} \left( x + \frac{4}{5} \right) = \frac{24}{25} \), we can follow these steps:
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Multiply both sides by \( \frac{11}{8} \) to isolate the term involving \( x \):
\[ x + \frac{4}{5} = \frac{24}{25} \cdot \frac{11}{8} \]
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Calculate the right-hand side:
\[ \frac{24 \cdot 11}{25 \cdot 8} = \frac{264}{200} = \frac{66}{50} = \frac{33}{25} \]
Thus, the equation becomes:
\[ x + \frac{4}{5} = \frac{33}{25} \]
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Now, subtract \( \frac{4}{5} \) from both sides. First, we need a common denominator for the fractions. The common denominator for 5 and 25 is 25. Convert \( \frac{4}{5} \) to a fraction with a denominator of 25:
\[ \frac{4}{5} = \frac{20}{25} \]
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Now subtract:
\[ x = \frac{33}{25} - \frac{20}{25} \]
\[ x = \frac{33 - 20}{25} = \frac{13}{25} \]
So, the solution is:
\[ \boxed{\frac{13}{25}} \]
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