Equations involving absolute values or square roots often
have extraneous answers.  (Answers that don't check in the
original equations, and thus are not solutions).  This
is such a case.  Let's go through the whole thing:

|16 - 3x| = 4x - 12

Case 1: when 16-3x equals the value of the right side.

16 - 3x = 4x - 12
    -7x = -28
      x = 4


Case 2: when 16-3x equals -1 times the value of the right side.

16 - 3x = -(4x - 12)
    -7x = -4x + 12
    -3x = 12
      x = -4

You are right except for one thing.  You must check absolute
value equations for extraneous solutions. If you check x=4,
you will find that it is a solution:

Checking x = 4,

   |16-3x| = 4x-12

 |16-3(4)| = 4(4)-12
 |16 - 12| = 16 - 12
       |4| = 4
         4 = 4

That checks.  So 4 is a solution.

But watch what happens when we try checking x = -4 in the
original equation:

   |16-3x| = 4x-12

|16-3(-4)| = 4(-4)-12
 |16 + 12| = -16 - 12
      |28| = -28
        28 = -28

So, as you see that does not check. So -4 must be discarded
since it is not a solution.

[Incidentally the absolute value bars are on your keyboard,
just above the ENTER key and below the Backspace key. Hold
down the shift and press the key for the backward slash \.
What's printed on the key looks like two small vertical dashes,
one above the other, but it types as |.

Edwin