To solve the inequality $|x - 2| \leq 9$, we proceed as follows:

Step 1: Definition of Absolute Value Inequality

The inequality $|x - 2| \leq 9$ means that the distance of $x - 2$ from 0 is less than or equal to 9. It can be rewritten as: $-9 \leq x - 2 \leq 9$

Step 2: Solve for $x$

Now, isolate $x$ by adding 2 to all parts of the inequality: $-9 + 2 \leq x - 2 + 2 \leq 9 + 2$ $-7 \leq x \leq 11$

Final Solution:

The solution is: $x \in [-7, 11]$ This means $x$ can take any value from $-7$ to $11$, inclusive.

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Related Questions:

1. How would the solution change if $|x - 2| > 9$?
2. What does the graph of $|x - 2| \leq 9$ look like on a number line?
3. How does the solution change if the inequality were $|x - 2| < 9$?
4. Can this inequality be solved using any other methods, such as squaring both sides?
5. What are some real-world examples where absolute value inequalities apply?

Tip:

When dealing with absolute value inequalities, always split them into two cases based on the definition of absolute value!