3<=$$ \left\vert x-2\right\vert $$$$ <=8 $$

To solve the compound inequality 

\[
3 \leq \left\vert x - 2 \right\vert \leq 8,
\]

we need to break it down into two parts based on the absolute value definition:

1. **Solve the inequality $ \left\vert x - 2 \right\vert \geq 3 $:**

   The absolute value $ \left\vert x - 2 \right\vert \geq 3 $ translates to two cases:

   \[
   x - 2 \geq 3 \quad \text{or} \quad x - 2 \leq -3
   \]

   Solving each:

   - $ x - 2 \geq 3 $ 
     \[
     x \geq 5
     \]

   - $ x - 2 \leq -3 $
     \[
     x \leq -1
     \]

   So, $ x \in (-\infty, -1] \cup [5, \infty) $.

2. **Solve the inequality $ \left\vert x - 2 \right\vert \leq 8 $:**

   The absolute value $ \left\vert x - 2 \right\vert \leq 8 $ translates to:

   \[
   -8 \leq x - 2 \leq 8
   \]

   Solving this:

   \[
   -8 + 2 \leq x \leq 8 + 2
   \]
   \[
   -6 \leq x \leq 10
   \]

   So, $ x \in [-6, 10] $.

### Step 3: Combine the results

The solution to the compound inequality requires finding the intersection of the two intervals $ (-\infty, -1] \cup [5, \infty) $ and $ [-6, 10] $.

- $ (-\infty, -1] $ intersects with $ [-6, 10] $ in $ [-6, -1] $.
- $ [5, \infty) $ intersects with $ [-6, 10] $ in $ [5, 10] $.

Thus, the final solution is:

\[
x \in [-6, -1] \cup [5, 10]
\]

This represents all the values of $ x $ that satisfy the original compound inequality.

Let me know if you want any more details or have any questions!

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Here are some related questions:

1. How do you solve compound inequalities with absolute values?
2. What is the graphical representation of the solution to this inequality?
3. Can you explain how absolute value affects the direction of inequalities?
4. What are some examples of real-world scenarios where such inequalities might be used?
5. How can you check your solution for absolute value inequalities?

**Tip:** When dealing with compound inequalities involving absolute values, always consider breaking them down into separate inequalities and solving each part before combining the results.

Learn how to solve the compound inequality 3 <= |x - 2| <= 8 with a detailed step-by-step solution. This problem involves inequalities and absolute value concepts. Suitable for advanced high school students.

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