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.

Learn how to solve the absolute value inequality |x - 6| ≥ 6 with a detailed step-by-step solution. This problem explores key algebraic concepts, including splitting inequalities into two cases and understanding absolute value properties. Suitable for students in Grades 8-10.

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