∣x−4∣>12

To solve the inequality, we need to consider two cases:

Case 1: (x - 4) > 12

First, we add 4 to both sides to isolate the x-term.
(x - 4) + 4 > 12 + 4
x > 16

So, the solution for this case is x > 16.

Case 2: -(x - 4) > 12

Next, we distribute the negative sign inside the absolute value.
-x + 4 > 12

To isolate the x-term, we subtract 4 from both sides.
(-x + 4) - 4 > 12 - 4
-x > 8

Since we have a negative coefficient for x, we need to multiply both sides by -1. Remember that when we multiply or divide an inequality by a negative number, we need to flip the inequality symbol.
(-1)(-x) < 8(-1)
x < -8

So, the solution for this case is x < -8.

Combining the solutions from both cases, we get:
x > 16 OR x < -8