1. **State the problem:** Solve the equation $$3^{2x - 1} + 1 = 2$$ for $x$. 2. **Isolate the exponential term:** Subtract 1 from both sides: $$3^{2x - 1} + 1 - 1 = 2 - 1$$ $$3^{2x - 1} = 1$$ 3. **Recall the property of exponents:** For any base $a > 0$ and $a \neq 1$, if $$a^y = 1$$ then $$y = 0$$. 4. **Apply the property:** Set the exponent equal to zero: $$2x - 1 = 0$$ 5. **Solve for $x$:** $$2x = 1$$ $$x = \frac{1}{2}$$ **Final answer:** $$x = \frac{1}{2}$$