Absolute Value Inequality 919413

1. **State the problem:** Solve the inequality $$|x + 9| - 3 \geq -2$$. 2. **Isolate the absolute value:** Add 3 to both sides: $$|x + 9| - 3 + 3 \geq -2 + 3$$ $$|x + 9| \geq 1$$ 3. **Recall the rule for absolute value inequalities:** For $$|A| \geq c$$ where $$c > 0$$, the solution is $$A \leq -c$$ or $$A \geq c$$. 4. **Apply the rule:** $$x + 9 \leq -1 \quad \text{or} \quad x + 9 \geq 1$$ 5. **Solve each inequality:** - For $$x + 9 \leq -1$$: $$x \leq -1 - 9$$ $$x \leq -10$$ - For $$x + 9 \geq 1$$: $$x \geq 1 - 9$$ $$x \geq -8$$ 6. **Combine the solution:** $$x \leq -10 \quad \text{or} \quad x \geq -8$$ 7. **Interpretation:** The solution set includes all $$x$$ less than or equal to $$-10$$ or greater than or equal to $$-8$$. **Final answer:** $$x \leq -10 \text{ or } x \geq -8$$