|z| means modulus of z. That is, whatever the sign (+ or -) of the value of z, |z| is the positive equivalent (or zero).
For example,
|3| = 3
|-6| = 6
|6| = 6
|0| = 0
Now if you had |y| = 1 there are two possible values for y.
Firstly, assume y is positive or zero (y >= 0). In this case, regardless of what y is, |y| = y (because |y| just asks for the positive equivalent of y and y is already positive).
You would then have |y| = y = 1. This is your first answer for |y| = 1
But you must also consider the possibility y is negative (y < 0). Since |y| converts y to a postive value, you can say that if y < 0 then |y| = -y (since if y is negative, -y will be positive).
Therefore, |y| = -y = 1, y = -1
Hence, if |y| = 1, then y = 1 or y = -1.
In relation to your question, you need to consider two possibilities:
If 12 - 3x >= 0 (positive)
Then |12 - 3x| = 12 - 3x = 9
3x = 12 - 9 = 3
x = 1
If 12 - 3x < 0 (negative)
Then |12 - 3x| = -(12 - 3x) = 9
-12 + 3x = 9
3x = 21
x = 7
So the other solution is x = 7.
Hope that helps (and sorry for being so long-winded).
crestu