Final answer:
The solution to the inequality -3|4-x| < -15 is x < -1 or x > 9, after considering the cases for the absolute value and reversing the inequality sign upon dividing by a negative number.
Step-by-step explanation:
The inequality in question is -3|4-x| < -15. First, we divide both sides of the inequality by -3, remembering that this will reverse the inequality sign since -3 is negative. This yields |4-x| > 5. Now, we solve for the absolute value inequality which means we have to consider two cases.
Case 1 (positive): 4-x > 5
1. Subtract 4 from both sides: -x > 1
2. Multiply both sides by -1 (inequality sign flips): x < -1
Case 2 (negative): 4-x < -5
1. Subtract 4 from both sides: -x < -9
2. Multiply both sides by -1 (inequality sign flips): x > 9
The solution to the inequality is therefore x < -1 or x > 9.