1. **State the problem:** Solve the equation $$\frac{r}{3} - \frac{r - 4}{3} = \frac{3}{2} - \frac{2r - 5}{6}$$ for $r$. 2. **Identify the common denominator:** The denominators are 3, 3, 2, and 6. The least common denominator (LCD) is 6. 3. **Multiply both sides by 6 to clear denominators:** $$6 \times \left(\frac{r}{3} - \frac{r - 4}{3}\right) = 6 \times \left(\frac{3}{2} - \frac{2r - 5}{6}\right)$$ 4. **Simplify each term:** $$6 \times \frac{r}{3} = 2r$$ $$6 \times \frac{r - 4}{3} = 2(r - 4)$$ $$6 \times \frac{3}{2} = 9$$ $$6 \times \frac{2r - 5}{6} = 2r - 5$$ 5. **Rewrite the equation:** $$2r - 2(r - 4) = 9 - (2r - 5)$$ 6. **Distribute:** $$2r - 2r + 8 = 9 - 2r + 5$$ 7. **Simplify both sides:** $$0r + 8 = 14 - 2r$$ 8. **Add $2r$ to both sides:** $$8 + 2r = 14$$ 9. **Subtract 8 from both sides:** $$2r = 14 - 8$$ $$2r = 6$$ 10. **Divide both sides by 2:** $$\cancel{2}r = \frac{6}{\cancel{2}}$$ $$r = 3$$ **Final answer:** $$r = 3$$