To solve the equation \( \frac{1}{2}v + 8 = 3 - \frac{1}{3}v \), we first want to eliminate the fractions by finding a common denominator. The least common multiple of 2 and 3 is 6.
Let's multiply every term in the equation by 6:
\[ 6 \left( \frac{1}{2}v \right) + 6 \cdot 8 = 6 \cdot 3 - 6 \left( \frac{1}{3}v \right) \]
This simplifies to:
\[ 3v + 48 = 18 - 2v \]
Next, we will add \( 2v \) to both sides to combine the \( v \) terms:
\[ 3v + 2v + 48 = 18 \]
This simplifies to:
\[ 5v + 48 = 18 \]
Now, subtract 48 from both sides:
\[ 5v = 18 - 48 \]
This simplifies to:
\[ 5v = -30 \]
Finally, divide by 5 to solve for \( v \):
\[ v = \frac{-30}{5} \]
Thus, we find:
\[ v = -6 \]
The solution is:
\[ \boxed{-6} \]