To solve the equation \( \frac{8}{9}(x + \frac{1}{2}) = \frac{32}{3} \), we can start by eliminating the fraction on the left side by multiplying both sides of the equation by \( \frac{9}{8} \):
\[ x + \frac{1}{2} = \frac{32}{3} \cdot \frac{9}{8} \]
Calculating the right side:
\[ \frac{32}{3} \cdot \frac{9}{8} = \frac{32 \cdot 9}{3 \cdot 8} = \frac{288}{24} = 12 \]
Thus, we have:
\[ x + \frac{1}{2} = 12 \]
Next, we need to isolate \( x \). We can do this by subtracting \( \frac{1}{2} \) from both sides:
\[ x = 12 - \frac{1}{2} \]
To subtract, we can convert \( 12 \) to a fraction with a denominator of 2:
\[ 12 = \frac{24}{2} \]
Then, we have:
\[ x = \frac{24}{2} - \frac{1}{2} = \frac{24 - 1}{2} = \frac{23}{2} \]
Therefore, the solution is:
\[ \boxed{\frac{23}{2}} \]