Final answer:
To solve the given compound inequalities, each absolute value inequality is split into two cases, resolved into a set of basic inequalities, and then the individual solutions are analyzed together to form a final solution set. Some of the given inequalities have no common solution as they contradict each other.
Step-by-step explanation:
Solving Compound Inequalities
To solve the compound inequality | x - 5 | > 7, we consider two cases: when (x - 5) is positive and when (x -5) is negative. This leads to two separate inequalities, x - 5 > 7 and -(x - 5) > 7, which simplify to x > 12 and x < -2, respectively.
The second inequality | 2x + 3 | < 12 also divides into two cases: 2x + 3 < 12 and -(2x + 3) < 12. Solving these gives -4.5 < x < 4.5.
In the third inequality, (| 4x - 1 | - 11) / -3 < 20, we first remove the fraction by multiplying both sides by -3, remembering to reverse the inequality sign, to get | 4x - 1 | - 11 > -60. Then, we add 11 to both sides to isolate the absolute value, yielding | 4x - 1 | > -49. Since an absolute value cannot be negative, this inequality doesn't provide any restrictions on x.
The last inequality -3 | x + 2 | < -9 after dividing by -3 and reversing the inequality sign becomes | x + 2 | > 3, which results in two inequalities: x + 2 > 3 and -(x + 2) > 3, giving us x > 1 and x < -5.
Combining all four inequalities, the solution for x must satisfy: x > 12, x < -2, -4.5 < x < 4.5, x > 1, and x < -5. However, some of these inequalities contradict each other and may not have a common solution set.