We are given the equation:

\[
|4x - 2| - 8 = 0
\]

### Step 1: Isolate the absolute value expression
Add 8 to both sides of the equation:

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

### Step 2: Solve for the two cases
Since the absolute value of a number is always non-negative, the equation can be split into two possible cases:

**Case 1:**
\[
4x - 2 = 8
\]

Solve for \( x \):
\[
4x = 10 \quad (\text{Add 2 to both sides})
\]
\[
x = \frac{10}{4} = 2.5
\]

**Case 2:**
\[
4x - 2 = -8
\]

Solve for \( x \):
\[
4x = -6 \quad (\text{Add 2 to both sides})
\]
\[
x = \frac{-6}{4} = -1.5
\]

### Step 3: Final solutions
The solutions are:

\[
x = 2.5 \quad \text{or} \quad x = -1.5
\]

Would you like further details on any of these steps?

Here are 5 related questions to explore:
1. What is the definition of absolute value?
2. How do you handle equations with multiple absolute value terms?
3. How can you graph the function \( y = |4x - 2| - 8 \)?
4. How would you solve \( |4x - 2| + 8 = 0 \)?
5. How do absolute value inequalities differ from absolute value equations?

**Tip:** Always remember to isolate the absolute value expression before splitting it into cases.

This step-by-step guide explains how to solve the absolute value equation |4x - 2| - 8 = 0. It covers key algebraic concepts, including absolute value properties and linear equations. The problem is ideal for students in Grades 7-9.

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