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The given inequality is:

\[
x - 2 \leq 3x + 4 \leq x + 14
\]

This is a compound inequality, and we can break it down into two parts:

1. \( x - 2 \leq 3x + 4 \)
2. \( 3x + 4 \leq x + 14 \)

### Solving the first part:

\[
x - 2 \leq 3x + 4
\]

- Subtract \(x\) from both sides:

\[
-2 \leq 2x + 4
\]

- Subtract 4 from both sides:

\[
-6 \leq 2x
\]

- Divide by 2:

\[
x \geq -3
\]

### Solving the second part:

\[
3x + 4 \leq x + 14
\]

- Subtract \(x\) from both sides:

\[
2x + 4 \leq 14
\]

- Subtract 4 from both sides:

\[
2x \leq 10
\]

- Divide by 2:

\[
x \leq 5
\]

### Final Solution:

Combining the two parts, we get:

\[
-3 \leq x \leq 5
\]

Thus, the solution is:

\[
x \in [-3, 5]
\]

Let me know if you need any further clarifications! Would you like more details or have any questions? 

Here are some related questions to further explore:

1. What happens if the inequality had been strict (\(<\) or \(>\)) instead of including equality?
2. Can you graph the solution \( -3 \leq x \leq 5 \) on a number line?
3. How would you solve a three-part inequality involving fractions, such as \( \frac{x}{2} \leq 3x - 5 \leq 4x + 1 \)?
4. What would happen if there was no solution to this compound inequality?
5. Can you set up a similar inequality with different variables and solve it?

**Tip:** Always remember to check both parts of a compound inequality separately and then combine the solutions!

Solve the compound inequality: x - 2 ≤ 3x + 4 ≤ x + 14

Learn how to solve the compound inequality x - 2 ≤ 3x + 4 ≤ x + 14 step-by-step. This problem explains how to break the inequality into parts, solve each, and combine the solutions. Suitable for students in Grades 9-10.

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