Let's simplify the equation step by step.

The equation is:

\[
-\frac{3}{2} \cdot (7 - \frac{1}{6}) + 4 \cdot \left( x - \frac{1}{2} \right) = 3
\]

First, simplify \( 7 - \frac{1}{6} \):

\[
7 = \frac{42}{6} \quad \Rightarrow \quad 7 - \frac{1}{6} = \frac{42}{6} - \frac{1}{6} = \frac{41}{6}
\]

Now substituting this back into the equation gives:

\[
-\frac{3}{2} \cdot \frac{41}{6} + 4 \left( x - \frac{1}{2} \right) = 3
\]

Now calculate \( -\frac{3}{2} \cdot \frac{41}{6} \):

\[
-\frac{3 \cdot 41}{2 \cdot 6} = -\frac{123}{12} = -\frac{41}{4}
\]

Now, substituting this value back into the equation:

\[
-\frac{41}{4} + 4 \left( x - \frac{1}{2} \right) = 3
\]

Distributing \( 4 \):

\[
-\frac{41}{4} + 4x - 2 = 3
\]

Now, combine terms on the left side:

\[
4x - 2 - \frac{41}{4} = 3
\]

Convert \(-2\) into a fraction with a denominator of 4:

\[
-2 = -\frac{8}{4}
\]

So, combining these gives:

\[
4x - \frac{8}{4} - \frac{41}{4} = 3 \quad \Rightarrow \quad 4x - \frac{49}{4} = 3
\]

Now, moving \(-\frac{49}{4}\) to the right side:

\[
4x = 3 + \frac{49}{4}
\]

Convert 3 into a fraction:

\[
3 = \frac{12}{4} \quad \Rightarrow \quad 4x = \frac{12}{4} + \frac{49}{4} = \frac{61}{4}
\]

Dividing both sides by 4:

\[
x = \frac{61}{16}
\]

So the solution to the equation is:

\[
x = \frac{61}{16}
\]