Absolute Inequalities 4F741C

1. **Problem:** Solve $|2s - 7| \geq -1$. Since absolute value is always non-negative, $|2s - 7| \geq -1$ is always true for all real $s$. **Answer:** $\boxed{\text{All real numbers}}$. --- 2. **Problem:** Solve $|5p + 2| < -4$. Absolute value is always $\geq 0$, so it can never be less than a negative number. **Answer:** $\boxed{\text{No solution}}$. --- 3. **Problem:** Solve $|6t - 7| - 8 \geq 3$. Step 1: Add 8 to both sides: $$|6t - 7| \geq 3 + 8$$ $$|6t - 7| \geq 11$$ Step 2: Use definition of absolute value inequality: $$6t - 7 \leq -11 \quad \text{or} \quad 6t - 7 \geq 11$$ Step 3: Solve each: For $6t - 7 \leq -11$: $$6t \leq -11 + 7$$ $$6t \leq -4$$ $$t \leq \frac{\cancel{6}t}{\cancel{6}} \leq \frac{-4}{6} = -\frac{2}{3}$$ For $6t - 7 \geq 11$: $$6t \geq 11 + 7$$ $$6t \geq 18$$ $$t \geq \frac{18}{6} = 3$$ **Answer:** $\boxed{t \leq -\frac{2}{3} \text{ or } t \geq 3}$. --- 4. **Problem:** Solve $3|14 - m| > 18$. Step 1: Divide both sides by 3: $$|14 - m| > \frac{18}{3}$$ $$|14 - m| > 6$$ Step 2: Absolute value inequality: $$14 - m < -6 \quad \text{or} \quad 14 - m > 6$$ Step 3: Solve each: For $14 - m < -6$: $$-m < -6 - 14$$ $$-m < -20$$ Multiply both sides by $-1$ (reverse inequality): $$m > 20$$ For $14 - m > 6$: $$-m > 6 - 14$$ $$-m > -8$$ Multiply both sides by $-1$ (reverse inequality): $$m < 8$$ **Answer:** $\boxed{m < 8 \text{ or } m > 20}$. --- 5. **Problem:** Solve $2|3w + 8| - 13 \leq -5$. Step 1: Add 13 to both sides: $$2|3w + 8| \leq -5 + 13$$ $$2|3w + 8| \leq 8$$ Step 2: Divide both sides by 2: $$|3w + 8| \leq 4$$ Step 3: Absolute value inequality: $$-4 \leq 3w + 8 \leq 4$$ Step 4: Subtract 8 from all parts: $$-4 - 8 \leq 3w \leq 4 - 8$$ $$-12 \leq 3w \leq -4$$ Step 5: Divide all parts by 3: $$\frac{-12}{3} \leq w \leq \frac{-4}{3}$$ $$-4 \leq w \leq -\frac{4}{3}$$ **Answer:** $\boxed{-4 \leq w \leq -\frac{4}{3}}$. --- 6. **Problem:** Solve $|h - 5| \leq 10$. Step 1: Absolute value inequality: $$-10 \leq h - 5 \leq 10$$ Step 2: Add 5 to all parts: $$-10 + 5 \leq h \leq 10 + 5$$ $$-5 \leq h \leq 15$$ **Answer:** $\boxed{-5 \leq h \leq 15}$. --- 7. **Problem:** Solve $|4c + 5| > 7$. Step 1: Absolute value inequality: $$4c + 5 < -7 \quad \text{or} \quad 4c + 5 > 7$$ Step 2: Solve each: For $4c + 5 < -7$: $$4c < -7 - 5$$ $$4c < -12$$ $$c < \frac{-12}{4} = -3$$ For $4c + 5 > 7$: $$4c > 7 - 5$$ $$4c > 2$$ $$c > \frac{2}{4} = \frac{1}{2}$$ **Answer:** $\boxed{c < -3 \text{ or } c > \frac{1}{2}}$. --- 8. **Problem:** Solve $|19 - 4n| < 5$. Step 1: Absolute value inequality: $$-5 < 19 - 4n < 5$$ Step 2: Subtract 19 from all parts: $$-5 - 19 < -4n < 5 - 19$$ $$-24 < -4n < -14$$ Step 3: Divide all parts by $-4$ (reverse inequalities): $$\frac{-24}{-4} > n > \frac{-14}{-4}$$ $$6 > n > 3.5$$ Rewrite: $$3.5 < n < 6$$ **Answer:** $\boxed{3.5 < n < 6}$. --- 9. **Problem:** Solve $|3j - 1| + 6 > 0$. Step 1: Subtract 6 from both sides: $$|3j - 1| > -6$$ Since absolute value is always $\geq 0$, it is always greater than $-6$. **Answer:** $\boxed{\text{All real numbers}}$. --- 10. **Problem:** Solve $-4|6b - 8| \leq 12$. Step 1: Divide both sides by $-4$ (reverse inequality): $$|6b - 8| \geq \frac{12}{-4} = -3$$ Since absolute value is always $\geq 0$, it is always $\geq -3$. **Answer:** $\boxed{\text{All real numbers}}$.