To solve the equation \( \frac{1}{2} (6x - 10) = 7x + 15 \), we'll start by eliminating the fraction by multiplying both sides of the equation by 2:
\[ 2 \cdot \frac{1}{2} (6x - 10) = 2 \cdot (7x + 15) \]
This simplifies to:
\[ 6x - 10 = 14x + 30 \]
Next, we will isolate \( x \) by moving all the terms involving \( x \) to one side and the constant terms to the other:
\[ 6x - 14x = 30 + 10 \]
This simplifies to:
\[ -8x = 40 \]
Now we divide both sides by -8:
\[ x = \frac{40}{-8} \]
This simplifies to:
\[ x = -5 \]
Therefore, the value of \( x \) is:
\[ \boxed{-5} \]