To solve the equation:
\[ \frac{8}{9}(x + 1.2) = \frac{32}{3} \]
we can start by eliminating the fraction. First, multiply both sides of the equation by \(9\) to clear the denominator:
\[ 9 \cdot \frac{8}{9}(x + 1.2) = 9 \cdot \frac{32}{3} \]
This simplifies to:
\[ 8(x + 1.2) = 96 \]
Next, divide both sides by \(8\):
\[ x + 1.2 = 12 \]
Now, subtract \(1.2\) from both sides to solve for \(x\):
\[ x = 12 - 1.2 \]
Calculating the right side gives:
\[ x = 10.8 \]
Thus, the solution for \(x\) is:
\[ \boxed{10.8} \]