To solve the equation |5x| = x - 12, we need to consider two cases: when 5x is positive and when 5x is negative.

Case 1: 5x is positive
If 5x is positive, then the equation becomes 5x = x - 12.
Simplifying this equation, we get 4x = -12.
Dividing both sides by 4, we have x = -3.

Case 2: 5x is negative
If 5x is negative, then the equation becomes -5x = x - 12.
Simplifying this equation, we get -6x = -12.
Dividing both sides by -6, we have x = 2.

Therefore, the solutions to the equation |5x| = x - 12 are x = -3 and x = 2.