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 by multiplying both sides of the equation by the denominator of the fraction, which is \( 11 \):
\[ 8 \left( x + \frac{4}{5} \right) = \frac{24}{55} \times 11 \]
Calculating the right side:
\[ \frac{24 \times 11}{55} = \frac{264}{55} \]
Now we simplify \( \frac{264}{55} \):
Finding the greatest common divisor, we see both values share no common factors besides 1, so \( \frac{264}{55} \) is already in its simplest form.
Now our equation is:
\[ 8 \left( x + \frac{4}{5} \right) = \frac{264}{55} \]
Next, divide both sides by \( 8 \):
\[ x + \frac{4}{5} = \frac{264}{55} \div 8 \]
To divide by 8, we can multiply by \( \frac{1}{8} \):
\[ x + \frac{4}{5} = \frac{264}{55} \times \frac{1}{8} = \frac{264}{440} \]
Simplifying \( \frac{264}{440} \):
Finding the GCD, which is \( 88 \): \[ \frac{264 \div 88}{440 \div 88} = \frac{3}{5} \]
So now we have:
\[ x + \frac{4}{5} = \frac{3}{5} \]
To isolate \( x \), subtract \( \frac{4}{5} \) from both sides:
\[ x = \frac{3}{5} - \frac{4}{5} \]
Calculating this:
\[ x = \frac{3 - 4}{5} = \frac{-1}{5} \]
Thus, the solution for \( x \) is:
\[ \boxed{-\frac{1}{5}} \]