1. **State the problem:** Solve the inequality $|x - 3| \leq 12$ for $x \in \mathbb{R}$. 2. **Recall the definition and rule for absolute value inequalities:** For any real number $a$ and positive number $b$, the inequality $|a| \leq b$ means $-b \leq a \leq b$. 3. **Apply this rule to the given inequality:** $$|x - 3| \leq 12 \implies -12 \leq x - 3 \leq 12$$ 4. **Solve the compound inequality:** Add 3 to all parts: $$-12 + 3 \leq x - 3 + 3 \leq 12 + 3$$ $$-9 \leq x \leq 15$$ 5. **Interpretation:** The solution set includes all real numbers $x$ such that $x$ is between $-9$ and $15$, inclusive. **Final answer:** $$\boxed{-9 \leq x \leq 15}$$