1. **State the problem:** Solve the equation $$\frac{x-3}{2} + \frac{x+2}{3} = 5$$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to combine terms or clear fractions by multiplying both sides by the least common denominator (LCD). 3. **Find the LCD:** The denominators are 2 and 3, so the LCD is 6. 4. **Multiply both sides by 6 to clear fractions:** $$6 \times \left(\frac{x-3}{2} + \frac{x+2}{3}\right) = 6 \times 5$$ 5. **Distribute multiplication:** $$6 \times \frac{x-3}{2} + 6 \times \frac{x+2}{3} = 30$$ 6. **Simplify each term:** $$3(x-3) + 2(x+2) = 30$$ 7. **Expand the parentheses:** $$3x - 9 + 2x + 4 = 30$$ 8. **Combine like terms:** $$5x - 5 = 30$$ 9. **Add 5 to both sides:** $$5x - 5 + 5 = 30 + 5$$ $$5x = 35$$ 10. **Divide both sides by 5:** $$\frac{\cancel{5}x}{\cancel{5}} = \frac{35}{5}$$ $$x = 7$$ **Final answer:** $$x = 7$$