1. **State the problem:** Solve the equation $$\frac{3x}{4} - 5 = \frac{x}{2} + 1$$ for $x$. 2. **Write down the equation:** $$\frac{3x}{4} - 5 = \frac{x}{2} + 1$$ 3. **Goal:** Isolate $x$ on one side. 4. **Eliminate fractions by multiplying both sides by the least common denominator (LCD), which is 4:** $$4 \times \left(\frac{3x}{4} - 5\right) = 4 \times \left(\frac{x}{2} + 1\right)$$ 5. **Distribute:** $$4 \times \frac{3x}{4} - 4 \times 5 = 4 \times \frac{x}{2} + 4 \times 1$$ 6. **Simplify each term:** $$\cancel{4} \times \frac{3x}{\cancel{4}} - 20 = 2x + 4$$ which simplifies to $$3x - 20 = 2x + 4$$ 7. **Subtract $2x$ from both sides to get all $x$ terms on one side:** $$3x - 2x - 20 = 2x - 2x + 4$$ which simplifies to $$x - 20 = 4$$ 8. **Add 20 to both sides to isolate $x$:** $$x - 20 + 20 = 4 + 20$$ which simplifies to $$x = 24$$ **Final answer:** $$x = 24$$