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!