1. **State the problem:** Solve the inequality $$439 \cdot 9^{5-|x+1|} \geq \frac{1}{81}$$. 2. **Rewrite the bases:** Note that $$9 = 3^2$$ and $$\frac{1}{81} = 81^{-1} = (3^4)^{-1} = 3^{-4}$$. 3. **Express the inequality with base 3:** $$439 \cdot (3^2)^{5-|x+1|} \geq 3^{-4}$$ which simplifies to $$439 \cdot 3^{2(5-|x+1|)} \geq 3^{-4}$$. 4. **Isolate the exponential term:** Divide both sides by 439: $$3^{2(5-|x+1|)} \geq \frac{3^{-4}}{439}$$. 5. **Simplify the right side:** $$\frac{3^{-4}}{439} = 3^{-4} \cdot \frac{1}{439}$$. 6. **Since 3 to any power is positive and 439 is positive, compare exponents by taking logarithm base 3:** Because $$3^{2(5-|x+1|)} \geq \frac{3^{-4}}{439}$$ and $$\frac{3^{-4}}{439} < 3^{-4}$$, we need to be careful. 7. **Estimate $$\frac{3^{-4}}{439}$$:** $$3^{-4} = \frac{1}{81} \approx 0.0123457$$ So, $$\frac{3^{-4}}{439} \approx \frac{0.0123457}{439} \approx 2.81 \times 10^{-5}$$. 8. **Rewrite inequality:** $$3^{2(5-|x+1|)} \geq 2.81 \times 10^{-5}$$. 9. **Take logarithm base 3:** $$2(5 - |x+1|) \geq \log_3(2.81 \times 10^{-5})$$. 10. **Calculate $$\log_3(2.81 \times 10^{-5})$$:** $$\log_3(2.81 \times 10^{-5}) = \log_3(2.81) + \log_3(10^{-5})$$ Using $$\log_3(2.81) \approx 0.88$$ and $$\log_3(10^{-5}) = -5 \log_3(10) \approx -5 \times 2.0959 = -10.4795$$, So, $$\log_3(2.81 \times 10^{-5}) \approx 0.88 - 10.4795 = -9.5995$$. 11. **Solve for $$|x+1|$$:** $$2(5 - |x+1|) \geq -9.5995$$ $$10 - 2|x+1| \geq -9.5995$$ 12. **Isolate $$|x+1|$$:** $$-2|x+1| \geq -19.5995$$ Divide both sides by -2 (remember to reverse inequality): $$|x+1| \leq \frac{19.5995}{2} = 9.79975$$ 13. **Interpret the absolute value inequality:** $$|x+1| \leq 9.79975$$ means $$-9.79975 \leq x+1 \leq 9.79975$$ 14. **Solve for $$x$$:** $$-10.79975 \leq x \leq 8.79975$$ **Final answer:** $$\boxed{-10.79975 \leq x \leq 8.79975}$$