Final answer:
To solve inequalities with absolute values, we need to consider different cases depending on the value inside the absolute value. We solve each case separately and combine the solutions. Some inequalities have no solution.
Step-by-step explanation:
Solve the inequalities:
- |b| < 10
To solve this inequality, we need to consider two cases: when b is positive and when b is negative. In both cases, |b| will be equal to b. So we have b < 10 and -b < 10. Therefore, the solution for this inequality is -10 < b < 10. - -8|w| < 6
To solve this inequality, we need to divide both sides by -8. However, since it's a negative number, the inequality sign will flip. So we have |w| > -6/8 which simplifies to |w| > -3/4. Since the absolute value of any number is always positive, the solution is w < -3/4 or w > 3/4. - 7(|z| + 2) < 56
We can solve this inequality by first dividing both sides by 7, which will give us |z| + 2 < 8. Then we subtract 2 from both sides to get |z| < 6. Since the absolute value of any number is always positive, the solution is -6 < z < 6. - |u| - 9 > -9
To solve this inequality, we add 9 to both sides to get |u| > 0. Since the absolute value of any number is always positive or 0, the solution is u ≠ 0. - |b| > 10
To solve this inequality, we need to consider two cases: when b is positive and when b is negative. In both cases, |b| will be equal to b. So we have b > 10 and -b > 10. Therefore, the solution for this inequality is b < -10 or b > 10. - -7(|h| + 1) > 28
To solve this inequality, we can start by dividing both sides by -7. However, since it's a negative number, the inequality sign will flip. So we have |h| + 1 < -28/7 which simplifies to |h| + 1 < -4. Then we subtract 1 from both sides to get |h| < -5. However, the absolute value of any number is always positive, so there is no solution for this inequality. - |2-3n| > 5
To solve this inequality, we need to consider two cases: when 2-3n is positive and when 2-3n is negative. In the first case, we have 2-3n > 5 which simplifies to -3n > 3, and then dividing both sides by -3 gives us n < -1. In the second case, we have -(2-3n) > 5 which simplifies to -2+3n > 5. Subtracting 3n from both sides gives us -2 > -5+3n, and then adding 5 to both sides gives us 3 > 3n, which simplifies to n < 1. Therefore, the solution for this inequality is n < -1 or n < 1. - 5+|a| > 13
To solve this inequality, we subtract 5 from both sides to get |a| > 8. Since the absolute value of any number is always positive, the solution is a < -8 or a > 8.