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The inequality provided is:

\[
|x + 10| \leq 3x - 2
\]

To solve this inequality, we need to split it into two cases based on the properties of absolute values. The absolute value function can be rewritten as two separate inequalities:

### Case 1: \( x + 10 \geq 0 \) (i.e., \( x \geq -10 \))
In this case, the inequality becomes:
\[
x + 10 \leq 3x - 2
\]

Subtract \(x\) from both sides:
\[
10 \leq 2x - 2
\]

Add 2 to both sides:
\[
12 \leq 2x
\]

Divide by 2:
\[
x \geq 6
\]

Thus, for \( x \geq -10 \), the solution is \( x \geq 6 \).

### Case 2: \( x + 10 < 0 \) (i.e., \( x < -10 \))
In this case, the absolute value inequality becomes:
\[
-(x + 10) \leq 3x - 2
\]

Simplify:
\[
-x - 10 \leq 3x - 2
\]

Add \(x\) to both sides:
\[
-10 \leq 4x - 2
\]

Add 2 to both sides:
\[
-8 \leq 4x
\]

Divide by 4:
\[
x \geq -2
\]

However, we have a contradiction here because \(x < -10\) in this case, and \(x \geq -2\) is not possible under the condition \(x < -10\). Therefore, there is no solution from this case.

### Final Answer:
The solution to the inequality is \( x \geq 6 \).

Do you have any questions or would you like more details?

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Here are 5 follow-up questions based on this inequality:

1. What are the key properties of absolute values when solving inequalities?
2. How do we approach cases where there is a contradiction, as seen in Case 2?
3. How would the solution change if the inequality was \( |x + 10| < 3x - 2 \) instead of \( \leq \)?
4. What happens if we modify the inequality to \( |x + 10| \geq 3x - 2 \)?
5. Can we solve this inequality graphically to verify the solution?

**Tip:** When solving absolute value inequalities, always check the conditions in both cases to avoid contradictions.

Learn how to solve the absolute value inequality |x + 10| ≤ 3x - 2. This problem involves breaking the inequality into two cases and solving for the variable x. The solution focuses on understanding the properties of absolute values and inequalities. Suitable for students in Grades 9-11.

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