To solve the equation

\[
-\frac{5}{6} + 3x = \frac{2}{3} - \frac{1}{2}x,
\]

we can follow these steps:

1. **Eliminate the fractions**: Multiply through by the least common multiple (LCM) of the denominators (which is 6):

\[
6 \left(-\frac{5}{6}\right) + 6(3x) = 6\left(\frac{2}{3}\right) - 6\left(\frac{1}{2}x\right).
\]

This simplifies to:

\[
-5 + 18x = 4 - 3x.
\]

2. **Combine like terms**: To isolate \(x\), we can move all terms involving \(x\) to one side and constant terms to the other:

\[
18x + 3x = 4 + 5.
\]

This simplifies to:

\[
21x = 9.
\]

3. **Solve for \(x\)**: Divide both sides by 21:

\[
x = \frac{9}{21} = \frac{3}{7}.
\]

Thus, the solution is

\[
\boxed{\frac{3}{7}}.
\]