1. **State the problem:** Solve the equation $$\frac{x-3}{2} + \frac{x+1}{3} = 5$$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to combine terms and eliminate fractions. 3. **Find the least common denominator (LCD):** The denominators are 2 and 3, so $$\text{LCD} = 6$$. 4. **Multiply both sides by the LCD to clear fractions:** $$6 \times \left( \frac{x-3}{2} + \frac{x+1}{3} \right) = 6 \times 5$$ 5. **Distribute multiplication:** $$6 \times \frac{x-3}{2} + 6 \times \frac{x+1}{3} = 30$$ 6. **Simplify each term:** $$3(x-3) + 2(x+1) = 30$$ 7. **Expand the parentheses:** $$3x - 9 + 2x + 2 = 30$$ 8. **Combine like terms:** $$5x - 7 = 30$$ 9. **Isolate the variable term:** $$5x = 30 + 7$$ $$5x = 37$$ 10. **Divide both sides by 5:** $$x = \frac{37}{5}$$ 11. **Final answer:** $$x = \frac{37}{5}$$ or $$7.4$$. This means the solution to the equation is $$x = \frac{37}{5}$$.