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.