Compound Inequalities Dbb742

1. **State the problem:** Solve the compound inequality $0 < -6 - 2x < 2$. 2. **Use the rule for compound inequalities:** We can solve it by doing the same operation to all three parts. Start with: $$0 < -6 - 2x < 2$$ 3. **Add $6$ to all three parts:** $$0+6 < -6 - 2x + 6 < 2+6$$ $$6 < -2x < 8$$ 4. **Divide all three parts by $-2$:** Because we divide by a negative number, the inequality signs must flip. $$\frac{6}{-2} > \frac{-2x}{-2} > \frac{8}{-2}$$ $$\cancel{\frac{6}{-2}} > \cancel{\frac{-2x}{-2}} > \cancel{\frac{8}{-2}}$$ $$-3 > x > -4$$ 5. **Rewrite in increasing order:** $$-4 < x < -3$$ 6. **Final answer:** The solution is $-4 < x < -3$.