To solve the equation \( \frac{8}{11}(x + \frac{4}{5}) = \frac{24}{55} \), we will first eliminate the fraction by multiplying both sides by the least common multiple (LCM) of the denominators. The LCM of 11 and 55 is 55.
- Multiply both sides by 55:
\[ 55 \cdot \frac{8}{11}(x + \frac{4}{5}) = 55 \cdot \frac{24}{55} \]
- Simplify the left side:
\[ 5 \cdot 8(x + \frac{4}{5}) = 24 \]
\[ 40(x + \frac{4}{5}) = 24 \]
- Divide both sides by 40:
\[ x + \frac{4}{5} = \frac{24}{40} \]
- Simplify the fraction on the right:
\[ \frac{24}{40} = \frac{3}{5} \]
Thus, we have:
\[ x + \frac{4}{5} = \frac{3}{5} \]
- Subtract \(\frac{4}{5}\) from both sides:
\[ x = \frac{3}{5} - \frac{4}{5} \]
- This simplifies to:
\[ x = -\frac{1}{5} \]
So, the solution is:
\[ \boxed{-\frac{1}{5}} \]
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