To solve the equation \( \frac{1}{2}\left(x + \frac{1}{6}\right) = \frac{19}{12} \), we first eliminate the fraction on the left side by multiplying both sides of the equation by 2:
\[ 2 \cdot \frac{1}{2}\left(x + \frac{1}{6}\right) = 2 \cdot \frac{19}{12} \]
This simplifies to:
\[ x + \frac{1}{6} = \frac{38}{12} \]
Next, we simplify \( \frac{38}{12} \):
\[ \frac{38}{12} = \frac{19}{6} \]
Now our equation is:
\[ x + \frac{1}{6} = \frac{19}{6} \]
To isolate \( x \), we subtract \( \frac{1}{6} \) from both sides:
\[ x = \frac{19}{6} - \frac{1}{6} \]
Since both fractions have the same denominator, we can directly subtract the numerators:
\[ x = \frac{19 - 1}{6} = \frac{18}{6} \]
Now, simplify \( \frac{18}{6} \):
\[ x = 3 \]
Thus, the solution is
\[ \boxed{3} \]