To solve for \( z \) in the equation
\[ \frac{2}{3}z = 2 + \frac{3}{2}z - \frac{1}{2}, \]
let's first simplify the right-hand side. We need a common denominator to combine the terms. The common denominator for 1 and 2 is 2, so we rewrite the terms:
\[ 2 = \frac{4}{2}, \]
Thus, we can rewrite the equation:
\[ \frac{2}{3}z = \frac{4}{2} + \frac{3}{2}z - \frac{1}{2}. \]
Now, combine the terms on the right side:
\[ \frac{4}{2} - \frac{1}{2} = \frac{3}{2}, \]
so now the equation looks like this:
\[ \frac{2}{3}z = \frac{3}{2} + \frac{3}{2}z. \]
Next, we want to isolate \( z \). To do this, we can subtract \( \frac{3}{2}z \) from both sides:
\[ \frac{2}{3}z - \frac{3}{2}z = \frac{3}{2}. \]
To combine the terms on the left side, we need a common denominator for \(\frac{2}{3}\) and \(\frac{3}{2}\). The common denominator is 6. Rewrite the fractions:
\[ \frac{2}{3}z = \frac{4}{6}z, \quad \frac{3}{2}z = \frac{9}{6}z. \]
Now substitute back into the equation:
\[ \frac{4}{6}z - \frac{9}{6}z = \frac{3}{2}. \]
Combine the left side:
\[ \left(\frac{4}{6} - \frac{9}{6}\right)z = \frac{3}{2}. \]
This gives:
\[ -\frac{5}{6}z = \frac{3}{2}. \]
To isolate \( z \), multiply both sides by the reciprocal of \(-\frac{5}{6}\):
\[ z = \frac{3}{2} \cdot -\frac{6}{5}. \]
Calculating the right side:
\[ z = -\frac{3 \cdot 6}{2 \cdot 5} = -\frac{18}{10} = -\frac{9}{5}. \]
Thus, the solution for \( z \) is
\[ \boxed{-\frac{9}{5}}. \]