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.