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We are tasked with solving the absolute value equation:

\[
|2x - 8| - 4 = 14
\]

### Step 1: Isolate the absolute value
To start, we move the \(-4\) to the other side of the equation by adding 4 to both sides:

\[
|2x - 8| = 14 + 4
\]

\[
|2x - 8| = 18
\]

### Step 2: Split the equation
The absolute value equation \( |2x - 8| = 18 \) can be split into two linear equations:

1. \( 2x - 8 = 18 \)
2. \( 2x - 8 = -18 \)

### Step 3: Solve both equations

#### First equation: \( 2x - 8 = 18 \)
\[
2x = 18 + 8
\]
\[
2x = 26
\]
\[
x = \frac{26}{2} = 13
\]

#### Second equation: \( 2x - 8 = -18 \)
\[
2x = -18 + 8
\]
\[
2x = -10
\]
\[
x = \frac{-10}{2} = -5
\]

### Step 4: Final Solution
The solutions to the equation are:
\[
x = 13 \quad \text{or} \quad x = -5
\]

Would you like further details or have any questions about this solution?

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Here are five related questions to explore this concept further:

1. What are the properties of absolute value functions that make splitting into two cases necessary?
2. How would you graph the absolute value function involved in this equation?
3. Can you solve a similar absolute value equation with different constants? For example, \( |3x + 2| - 5 = 9 \)?
4. How does the solution change if the right-hand side of the absolute value equation is negative?
5. What is the geometric interpretation of absolute value equations?

**Tip:** Always remember that the absolute value represents the distance from zero, so any equation involving absolute values typically has two cases to consider.

Solve the absolute value equation |2x - 8| - 4 = 14.

This page provides a detailed solution to the absolute value equation |2x - 8| - 4 = 14. Learn how to isolate the absolute value and solve for two cases. This problem introduces key algebraic concepts and is ideal for students in Grades 8-10.

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