Let's solve the equation \( |6 - |2x - 4|| = 2 \).

### Step 1: Break down the absolute values

We need to consider the absolute values separately. The outermost absolute value is applied to the expression \( 6 - |2x - 4| \).

Start by solving the inside expression:

\[
|2x - 4|
\]

### Step 2: Case 1 - \( 6 - |2x - 4| = 2 \)

For the first case, remove the absolute value sign and directly solve the equation:

\[
6 - |2x - 4| = 2
\]

Subtract 6 from both sides:

\[
-|2x - 4| = -4
\]

Multiply through by -1:

\[
|2x - 4| = 4
\]

Now, solve the absolute value equation:

- Case A: \( 2x - 4 = 4 \)
  \[
  2x = 8 \quad \Rightarrow \quad x = 4
  \]

- Case B: \( 2x - 4 = -4 \)
  \[
  2x = 0 \quad \Rightarrow \quad x = 0
  \]

### Step 3: Case 2 - \( 6 - |2x - 4| = -2 \)

For the second case, consider the scenario where:

\[
6 - |2x - 4| = -2
\]

Subtract 6 from both sides:

\[
-|2x - 4| = -8
\]

Multiply through by -1:

\[
|2x - 4| = 8
\]

Now, solve the absolute value equation:

- Case A: \( 2x - 4 = 8 \)
  \[
  2x = 12 \quad \Rightarrow \quad x = 6
  \]

- Case B: \( 2x - 4 = -8 \)
  \[
  2x = -4 \quad \Rightarrow \quad x = -2
  \]

### Final Solution

The possible values of \( x \) are:
\[
x = 4, 0, 6, -2
\]

If you want more details on how absolute value equations work or further explanation, let me know.

### Follow-up Questions

1. How do you handle absolute value inequalities?
2. Can absolute values be negative?
3. What are some real-life applications of absolute values?
4. How would the solution change if the right-hand side were negative?
5. What happens if there are nested absolute values with different coefficients?

### Tip:
When solving absolute value equations, always consider both positive and negative cases for the expression inside the absolute value.

Learn how to solve the absolute value equation |6 - |2x - 4|| = 2 step-by-step with detailed explanations. This problem involves absolute value equations and requires careful consideration of different cases. Suitable for advanced high school students and beyond.

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