1. **State the problem:** Solve for $x$ in the equation $$\frac{3^{2x-1}}{3^{x+1}} = 9.$$\n\n2. **Recall the properties of exponents:** When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}.$$ Also, express constants as powers of the same base if possible. Here, $9 = 3^2$.\n\n3. **Apply the division rule:**\n$$\frac{3^{2x-1}}{3^{x+1}} = 3^{(2x-1)-(x+1)} = 3^{2x-1-x-1} = 3^{x-2}.$$\n\n4. **Rewrite the equation:**\n$$3^{x-2} = 3^2.$$\n\n5. **Since the bases are equal, set the exponents equal:**\n$$x - 2 = 2.$$\n\n6. **Solve for $x$:**\n$$x = 2 + 2 = 4.$$\n\n**Final answer:** $$x = 4.$$