1. **State the problem:** Solve the inequality $$|x+1| - 4 > 9$$. 2. **Isolate the absolute value:** Add 4 to both sides: $$|x+1| - 4 + 4 > 9 + 4$$ $$|x+1| > 13$$ 3. **Recall the rule for absolute value inequalities:** For $$|A| > B$$ where $$B > 0$$, the solution is $$A < -B$$ or $$A > B$$. 4. **Apply the rule:** $$x + 1 < -13 \quad \text{or} \quad x + 1 > 13$$ 5. **Solve each inequality:** - For $$x + 1 < -13$$: $$x < -13 - 1$$ $$x < -14$$ - For $$x + 1 > 13$$: $$x > 13 - 1$$ $$x > 12$$ 6. **Write the final solution:** $$x < -14 \quad \text{or} \quad x > 12$$ This means $$x$$ is any number less than $$-14$$ or greater than $$12$$.