To solve the equation \( 3(4x - 2) = 1 \), we start by distributing the 3 on the left side:
\[ 3 \cdot 4x - 3 \cdot 2 = 1 \]
This simplifies to:
\[ 12x - 6 = 1 \]
Next, we can isolate \( 12x \) by adding 6 to both sides of the equation:
\[ 12x - 6 + 6 = 1 + 6 \]
This simplifies to:
\[ 12x = 7 \]
Now, to find \( x \), we divide both sides by 12:
\[ x = \frac{7}{12} \]
Thus, the value of \( x \) is:
\[ \boxed{\frac{7}{12}} \]