We are tasked with solving the inequality:

$|x - 2| \geq 9$

Step 1: Understanding the Absolute Value Inequality

An inequality involving absolute value can be split into two separate inequalities. For $|x - 2| \geq 9$, this means:

1. $x - 2 \geq 9$
2. $x - 2 \leq -9$

Step 2: Solve Each Inequality

1. $x - 2 \geq 9$

   • Add 2 to both sides: $x \geq 11$

2. $x - 2 \leq -9$

   • Add 2 to both sides: $x \leq -7$

Step 3: Combine the Solutions

The solution is $x \geq 11$ or $x \leq -7$. This describes two separate intervals: one where $x$ is greater than or equal to 11, and one where $x$ is less than or equal to -7.

Conclusion:

The correct answer is: $\boxed{-x \leq -7 \text{ or } x \geq 11}$ This corresponds to the second option in the list.

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Do you need more details or explanations? Here are five related questions to consider:

1. What is the general approach for solving absolute value inequalities?
2. Can we apply this method to absolute value inequalities involving quadratic expressions?
3. How do you graph the solution of an absolute value inequality on a number line?
4. What are the differences between strict inequalities and non-strict inequalities in absolute value problems?
5. How does solving absolute value equations differ from solving inequalities?

Tip: When solving absolute value inequalities, always remember to split the inequality into two cases: one where the expression inside is positive, and one where it's negative!