Absolute Value Inequality 232383

1. **State the problem:** Solve the absolute value inequality $$|3x - 6| \leq 12$$. 2. **Recall the rule for absolute value inequalities:** For $$|A| \leq B$$ where $$B \geq 0$$, the inequality is equivalent to $$-B \leq A \leq B$$. 3. **Apply the rule:** Here, $$A = 3x - 6$$ and $$B = 12$$, so $$-12 \leq 3x - 6 \leq 12$$ 4. **Solve the compound inequality:** Add 6 to all parts: $$-12 + 6 \leq 3x - 6 + 6 \leq 12 + 6$$ $$-6 \leq 3x \leq 18$$ 5. **Divide all parts by 3:** $$\frac{-6}{3} \leq \frac{3x}{3} \leq \frac{18}{3}$$ Show cancellation: $$\cancel{3} \times \frac{-6}{\cancel{3}} \leq x \leq \cancel{3} \times \frac{18}{\cancel{3}}$$ Simplify: $$-2 \leq x \leq 6$$ 6. **Interpret the solution:** The solution set is all $$x$$ such that $$-2 \leq x \leq 6$$. 7. **Match with given options:** This corresponds to option (c) $$-2 \leq x \leq 6$$. **Final answer:** (c) $$-2 \leq x \leq 6$$.