Final answer:
A compound inequality that represents the solution set (-7, -3] U [5, 9] is -9 ≤ x < -3 or 5 < x ≤ 9, breaking it into two parts according to the distance from zero on a number line.
Step-by-step explanation:
To create a compound inequality with an absolute value that has a solution set of (-7, -3] U [5, 9], we need to think about what an absolute value inequality represents. An absolute value measures the distance of a number from zero on a number line, irrespective of the direction. Therefore, we are looking for the set of all x values whose distance from 0 is greater than 4 and less than or equal to 7.
The inequality can be written as: -7 < |x| ≤ 9. This inequality can be broken into two parts since the absolute value of x can be both positive and negative: -9 ≤ x ≤ -4 or 4 < x ≤ 9. To combine these, we omit the overlapping parts and use the union of the solution sets:
-9 ≤ x < -3 or 5 < x ≤ 9