To solve the inequality \(|x - 8| \geq 1\), follow these steps:

1. **Interpret the absolute value**:
   The inequality \(|x - 8| \geq 1\) means that the distance between \(x\) and 8 is at least 1. This can be split into two cases:
   \[
   x - 8 \geq 1 \quad \text{or} \quad x - 8 \leq -1
   \]

2. **Solve each case**:

   - For the first case:
     \[
     x - 8 \geq 1
     \]
     Add 8 to both sides:
     \[
     x \geq 9
     \]

   - For the second case:
     \[
     x - 8 \leq -1
     \]
     Add 8 to both sides:
     \[
     x \leq 7
     \]

3. **Combine the solutions**:
   The solution to the inequality is \(x \geq 9\) or \(x \leq 7\).

Thus, the solution is \(\boxed{x \leq 7 \text{ or } x \geq 9}\).

Would you like further details or have any questions? Here are some related questions:

1. How do you interpret absolute value inequalities?
2. What is the difference between \(|x - a| \geq b\) and \(|x - a| \leq b\)?
3. How do you graph the solution set for a disjoint inequality?
4. Can you explain how absolute value inequalities relate to distance on a number line?
5. How do you solve \(|ax + b| \geq c\) for general values of \(a\), \(b\), and \(c\)?

**Tip:** For absolute value inequalities like \(|x - a| \geq b\), the solution is usually in two parts: either \(x\) is at least \(b\) units away from \(a\), which means \(x \geq a + b\) or \(x \leq a - b\).

Learn how to solve the absolute value inequality |x - 8| >= 1 with detailed steps and explanations. This problem covers absolute value concepts, inequalities, and their applications on the number line. Suitable for students in Grades 7-9.

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