1. **Problem:** Solve the equation $\frac{1}{2}x + \frac{3}{2}(x+1) - \frac{1}{4} = 5$. 2. **Use the distributive property first.** $$\frac{1}{2}x + \frac{3}{2}x + \frac{3}{2} - \frac{1}{4} = 5$$ 3. **Combine like terms.** $$\frac{1}{2}x + \frac{3}{2}x = \frac{4}{2}x = 2x$$ So the equation becomes $$2x + \frac{3}{2} - \frac{1}{4} = 5$$ 4. **Simplify the fractions.** $$\frac{3}{2} - \frac{1}{4} = \frac{6}{4} - \frac{1}{4} = \frac{5}{4}$$ So now we have $$2x + \frac{5}{4} = 5$$ 5. **Subtract $\frac{5}{4}$ from both sides.** $$2x = 5 - \frac{5}{4}$$ Rewrite $5$ with denominator $4$: $$2x = \frac{20}{4} - \frac{5}{4}$$ $$2x = \frac{15}{4}$$ 6. **Divide both sides by $2$.** $$x = \frac{15}{4} \div 2$$ $$x = \frac{15}{4} \times \frac{1}{2}$$ $$x = \frac{15}{8}$$ 7. **Final answer:** The solution is **$\frac{15}{8}$**, which is **C**.