The inequality you provided is:

\[
x - 8 > |x|
\]

Let's solve this step by step.

### Step 1: Consider the cases for the absolute value.
The absolute value \( |x| \) can be split into two cases depending on whether \( x \) is positive or negative.

- Case 1: \( x \geq 0 \), so \( |x| = x \)
- Case 2: \( x < 0 \), so \( |x| = -x \)

### Case 1: \( x \geq 0 \)
The inequality becomes:

\[
x - 8 > x
\]

Subtracting \( x \) from both sides:

\[
-8 > 0
\]

This is **false**, meaning there is no solution in this case.

### Case 2: \( x < 0 \)
In this case, the inequality becomes:

\[
x - 8 > -x
\]

Add \( x \) to both sides:

\[
2x - 8 > 0
\]

Now, solve for \( x \):

\[
2x > 8 \implies x > 4
\]

However, this contradicts the assumption that \( x < 0 \), so there is no solution in this case either.

### Final Answer:
There is no solution to the inequality \( x - 8 > |x| \).

Would you like a more detailed explanation of any of these steps, or have any questions?

Here are some related questions:
1. How do you solve inequalities with absolute values in general?
2. What happens if we change \(x - 8 \geq |x|\)?
3. How does this inequality behave graphically?
4. What are some real-world applications of absolute value inequalities?
5. How do inequalities change when absolute values are inside other functions?

**Tip:** Always remember to check for contradictions when solving absolute value inequalities by testing both cases.

This problem involves solving the inequality X - 8 > |x| by analyzing different cases for absolute values. Learn the step-by-step solution, which applies key algebraic concepts like inequalities and absolute value properties. Suitable for high school students.

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