1. **State the problem:** Solve the inequality $$0.25^x \leq 8^{x+1}$$. 2. **Rewrite the inequality using properties of exponents:** $$0.25^x \leq 8^x \times 8$$ 3. **Express the bases as powers of 2:** - Note that $$0.25 = \frac{1}{4} = 4^{-1} = (2^2)^{-1} = 2^{-2}$$ - Also, $$8 = 2^3$$ So the inequality becomes: $$\left(2^{-2}\right)^x \leq \left(2^3\right)^x \times 2^3$$ 4. **Simplify the powers:** $$2^{-2x} \leq 2^{3x} \times 2^3 = 2^{3x + 3}$$ 5. **Since the base 2 is positive and greater than 1, we can compare exponents:** $$-2x \leq 3x + 3$$ 6. **Solve the inequality for x:** $$-2x - 3x \leq 3$$ $$-5x \leq 3$$ 7. **Divide both sides by -5, remembering to reverse the inequality sign because we divide by a negative number:** $$x \geq \frac{3}{-5}$$ $$x \geq -\frac{3}{5}$$ **Final answer:** $$\boxed{x \geq -\frac{3}{5}}$$