Let's solve the equation $|3x - 6| - 9 = -3$.

Step 1: Isolate the absolute value expression

Add 9 to both sides of the equation to isolate the absolute value: $|3x - 6| = -3 + 9$ $|3x - 6| = 6$

Step 2: Solve the absolute value equation

By definition, $|A| = B$ means $A = B$ or $A = -B$. So: $3x - 6 = 6 \quad \text{or} \quad 3x - 6 = -6$

Case 1: $3x - 6 = 6$

Add 6 to both sides: $3x = 12$ Divide by 3: $x = 4$

Case 2: $3x - 6 = -6$

Add 6 to both sides: $3x = 0$ Divide by 3: $x = 0$

Step 3: Final answer

The solutions are: $x = 4 \quad \text{or} \quad x = 0$

Would you like further details or have any questions?

Here are some related questions:

1. What happens if the absolute value results in a negative number?
2. Can you solve absolute value inequalities?
3. How do you graph absolute value functions?
4. What is the general form of an absolute value equation?
5. How does the solution change if the equation is $|3x - 6| + 9 = 3$?

Tip: Always isolate the absolute value before solving the equation.