To solve the equation
\[ \frac{8}{11} (x + 4.5) = \frac{24}{25}, \]
we will first eliminate the fraction by multiplying both sides by the least common multiple of the denominators, which is \( 275 \) (since \( 275 = 11 \times 25 \)).
Multiplying both sides by \( 275 \):
\[ 275 \cdot \frac{8}{11} (x + 4.5) = 275 \cdot \frac{24}{25}. \]
This simplifies to:
\[ 25 \cdot 8 (x + 4.5) = 11 \cdot 24. \]
Calculating each side:
\[ 200 (x + 4.5) = 264. \]
Next, we can divide both sides by 200:
\[ x + 4.5 = \frac{264}{200}. \]
Simplifying \( \frac{264}{200} \):
\[ \frac{264 \div 8}{200 \div 8} = \frac{33}{25} = 1.32. \]
Thus, we have:
\[ x + 4.5 = 1.32. \]
Next, we isolate \( x \) by subtracting \( 4.5 \) from both sides:
\[ x = 1.32 - 4.5. \]
Calculating the right-hand side:
\[ x = 1.32 - 4.50 = -3.18. \]
Thus, the solution to the equation is
\[ \boxed{-3.18}. \]