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