The inequality \( |x - 4| > 8 \) means that the absolute difference between \(x\) and 4 is greater than 8. To solve this inequality, we need to consider the definition of absolute value, which gives us two possible cases:

### Case 1: \( x - 4 > 8 \)
Solve for \(x\):
\[
x - 4 > 8
\]
\[
x > 12
\]

### Case 2: \( x - 4 < -8 \)
Solve for \(x\):
\[
x - 4 < -8
\]
\[
x < -4
\]

### Final Solution:
Combining both cases, the solution to the inequality is:
\[
x > 12 \quad \text{or} \quad x < -4
\]
In interval notation, the solution is:
\[
(-\infty, -4) \cup (12, \infty)
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the solution to \( |x + 3| \leq 5 \)?
2. How do you solve \( |x - 2| = 7 \)?
3. What is the general form of solving absolute value inequalities?
4. What are the steps to graph the solution \( |x - 4| > 8 \)?
5. How can we solve compound inequalities involving absolute values?

**Tip:** Always split absolute value inequalities into two cases—one for the positive and one for the negative scenario.

Learn how to solve the absolute value inequality ∣x−4∣>8 with a detailed step-by-step solution. This problem covers key algebraic concepts, such as solving absolute value inequalities and applying the absolute value inequality theorem. Suitable for students in Grades 9-11.

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