1. **State the problem:** Solve the equation $$\frac{3x}{4} - \frac{1}{4}Cx - 20 = \frac{x}{4} + 32$$ where $C$ is a constant. 2. **Rewrite the equation:** Move all terms to one side to isolate $x$ terms and constants. 3. **Combine like terms:** Group $x$ terms and constants separately. 4. **Isolate $x$:** Use algebraic operations to solve for $x$. 5. **Show intermediate steps:** $$\frac{3x}{4} - \frac{1}{4}Cx - 20 = \frac{x}{4} + 32$$ Subtract $\frac{x}{4}$ from both sides: $$\frac{3x}{4} - \frac{1}{4}Cx - \cancel{\frac{x}{4}} - 20 = \cancel{\frac{x}{4}} + 32 - \frac{x}{4}$$ Simplify: $$\frac{3x}{4} - \frac{1}{4}Cx - \frac{x}{4} - 20 = 32$$ Combine $x$ terms: $$\left(\frac{3}{4} - \frac{1}{4}C - \frac{1}{4}\right)x - 20 = 32$$ Simplify coefficients: $$\left(\frac{3}{4} - \frac{1}{4} - \frac{1}{4}C\right)x - 20 = 32$$ $$\left(\frac{2}{4} - \frac{1}{4}C\right)x - 20 = 32$$ $$\left(\frac{1}{2} - \frac{C}{4}\right)x - 20 = 32$$ Add 20 to both sides: $$\left(\frac{1}{2} - \frac{C}{4}\right)x = 32 + 20$$ $$\left(\frac{1}{2} - \frac{C}{4}\right)x = 52$$ Divide both sides by $\left(\frac{1}{2} - \frac{C}{4}\right)$: $$x = \frac{52}{\frac{1}{2} - \frac{C}{4}}$$ Rewrite denominator with common denominator 4: $$x = \frac{52}{\frac{2}{4} - \frac{C}{4}} = \frac{52}{\frac{2 - C}{4}}$$ Invert and multiply: $$x = 52 \times \frac{4}{2 - C} = \frac{208}{2 - C}$$ **Final answer:** $$x = \frac{208}{2 - C}$$ This solution expresses $x$ in terms of the constant $C$.