1. **State the problem:** Solve for $x$ in the equation $$3^{x+1} = 5^{2-x}$$ and find the value of $x$ step by step. 2. **Recall the formula and rules:** To solve equations with variables in exponents, we use logarithms. The key property is $$a^b = c \implies b = \log_a(c)$$ or equivalently, take the natural logarithm (ln) of both sides. 3. **Take natural logarithm of both sides:** $$\ln\left(3^{x+1}\right) = \ln\left(5^{2-x}\right)$$ 4. **Use logarithm power rule:** $$(x+1) \ln(3) = (2 - x) \ln(5)$$ 5. **Distribute logarithms:** $$x \ln(3) + \ln(3) = 2 \ln(5) - x \ln(5)$$ 6. **Group terms with $x$ on one side:** $$x \ln(3) + x \ln(5) = 2 \ln(5) - \ln(3)$$ 7. **Factor out $x$:** $$x (\ln(3) + \ln(5)) = 2 \ln(5) - \ln(3)$$ 8. **Solve for $x$:** $$x = \frac{2 \ln(5) - \ln(3)}{\ln(3) + \ln(5)}$$ 9. **Calculate numerical values:** $$\ln(3) \approx 1.0986, \quad \ln(5) \approx 1.6094$$ 10. **Substitute and simplify:** $$x = \frac{2(1.6094) - 1.0986}{1.0986 + 1.6094} = \frac{3.2188 - 1.0986}{2.708} = \frac{2.1202}{2.708}$$ 11. **Final value:** $$x \approx 0.7833$$ 12. **Round to nearest hundredth:** $$x \approx 0.78$$ **Answer:** $$x \approx 0.78$$