The inequality you are being asked to solve is:

$|x - 4| > 8$

To solve this absolute value inequality, follow these steps:

Step 1: Break it into two inequalities.

The expression $|x - 4| > 8$ means that the quantity inside the absolute value, $x - 4$, must be either greater than 8 or less than -8. This gives us two inequalities:

$x - 4 > 8 \quad \text{or} \quad x - 4 < -8$

Step 2: Solve each inequality.

Inequality 1: $x - 4 > 8$

Add 4 to both sides: $x > 12$

Inequality 2: $x - 4 < -8$

Add 4 to both sides: $x < -4$

Step 3: Combine the solution.

The solution to the inequality is that $x$ is either greater than 12 or less than -4. In interval notation, the solution is:

$(-\infty, -4) \cup (12, \infty)$

This represents all real numbers less than -4 or greater than 12.

Do you want further details or have any questions?
Here are 5 related questions to expand your understanding:

1. What is the difference between "greater than" and "greater than or equal to" in absolute value inequalities?
2. How would the solution change if the inequality was $|x - 4| \geq 8$?
3. How can absolute value inequalities be represented on a number line?
4. What are the key steps to follow when solving inequalities algebraically?
5. Can absolute value inequalities have no solution? In which cases?

Tip:

When solving absolute value inequalities, always remember to split the equation into two cases and then solve them individually.