We are given the inequality:

$|x - 6| \geq 6$

To solve this absolute value inequality, we split it into two cases based on the definition of absolute value:

1. $x - 6 \geq 6$
2. $-(x - 6) \geq 6$ (or equivalently, $x - 6 \leq -6$)

Case 1: $x - 6 \geq 6$

$x \geq 12$

Case 2: $x - 6 \leq -6$

$x \leq 0$

Final Solution

The solution is the union of these two cases: $x \leq 0 \quad \text{or} \quad x \geq 12$ In interval notation, this is: $(-\infty, 0] \cup [12, \infty)$

Would you like more details or have any questions about this?

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Here are 5 related questions to further explore this topic:

1. How do you solve an absolute value equation?
2. What is the graphical representation of the solution $|x - 6| \geq 6$?
3. How does absolute value affect inequalities?
4. What is the difference between strict and non-strict inequalities?
5. How would the solution change if it were $|x - 6| > 6$?

Tip: When solving absolute value inequalities, always remember to split the inequality into two cases based on the absolute value definition.