1. **State the problem:** Solve the inequality $4^{x+3} \geq 4^x$. 2. **Recall the properties of exponents:** For any base $a > 0$ and $a \neq 1$, if $a^m \geq a^n$, then $m \geq n$ because the exponential function is strictly increasing. 3. **Apply the property:** Since the base is 4 (which is greater than 1), we can compare the exponents directly: $$x + 3 \geq x$$ 4. **Simplify the inequality:** $$x + 3 \geq x$$ Subtract $x$ from both sides: $$\cancel{x} + 3 \geq \cancel{x}$$ which simplifies to: $$3 \geq 0$$ 5. **Interpret the result:** The inequality $3 \geq 0$ is always true. 6. **Conclusion:** Since the inequality holds for all real $x$, the solution set is all real numbers. **Final answer:** $$\boxed{(-\infty, \infty)}$$