ANSWER: x=5/3, -1. Good luck. Hope this suffices.
Explanation for x=-1: |9x-3|=12. Lets take the absolute value brackets away for a sec. Add 3 to both sides 9x-3+3=-12+3, simplify: 9x=-9, divide both sides by nine:9x/9=-9/9. And that means that x=-1.
Explanation for x=5/3: |9x-3|=12. Add 3. 9x-3+3=12+3. Simplify: 9x=15. Divide both sides by 9. 9x/9=15/9. Simplify: x=5/3