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 $x \leq a - b$.