1. **State the problem:** Find the value of $z$ that satisfies the equation $$\frac{1}{4}(2 - z) + \frac{3}{2} = 3(z - 2).$$ 2. **Write the equation:** $$\frac{1}{4}(2 - z) + \frac{3}{2} = 3(z - 2).$$ 3. **Distribute the fractions:** $$\frac{1}{4} \times 2 - \frac{1}{4} \times z + \frac{3}{2} = 3z - 6.$$ This simplifies to $$\frac{2}{4} - \frac{z}{4} + \frac{3}{2} = 3z - 6.$$ 4. **Simplify fractions:** $$\frac{1}{2} - \frac{z}{4} + \frac{3}{2} = 3z - 6.$$ 5. **Combine like terms on the left:** $$\left(\frac{1}{2} + \frac{3}{2}\right) - \frac{z}{4} = 3z - 6,$$ which is $$2 - \frac{z}{4} = 3z - 6.$$ 6. **Add $\frac{z}{4}$ to both sides:** $$2 = 3z - 6 + \frac{z}{4}.$$ 7. **Combine like terms on the right:** $$3z + \frac{z}{4} - 6 = 2.$$ Convert $3z$ to $\frac{12z}{4}$ to add fractions: $$\frac{12z}{4} + \frac{z}{4} - 6 = 2,$$ which is $$\frac{13z}{4} - 6 = 2.$$ 8. **Add 6 to both sides:** $$\frac{13z}{4} = 8.$$ 9. **Multiply both sides by 4:** $$\cancel{4} \times \frac{13z}{\cancel{4}} = 8 \times 4,$$ which simplifies to $$13z = 32.$$ 10. **Divide both sides by 13:** $$z = \frac{32}{13}.$$ **Final answer:** $$z = \frac{32}{13}.$$