Solve the inequality: | 2x | ≤ 6   and tell me what that looks like on a number line. 

The inequality you provided is:

\[
|2x| \leq 6
\]

### Step 1: Break it into two inequalities

For absolute value inequalities, we can break it into two parts. The inequality \( |2x| \leq 6 \) translates into:

\[
-6 \leq 2x \leq 6
\]

### Step 2: Solve for \(x\)

To solve for \(x\), divide the entire inequality by 2:

\[
\frac{-6}{2} \leq x \leq \frac{6}{2}
\]

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

### Final solution:

The solution to the inequality is:

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

### Number Line Representation:

On a number line, this would be represented as a closed interval from \(-3\) to \(3\), meaning that both \(-3\) and \(3\) are included in the solution. To visualize it:

- Draw a line with numbers.
- Mark \(-3\) and \(3\) with solid dots (since they are included).
- Shade the region between \(-3\) and \(3\), including the endpoints.

Would you like me to draw this number line for you, or explain any part further?

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Here are 5 related questions:
1. How would the solution change if the inequality was \( |2x| > 6 \)?
2. How do we solve absolute value inequalities with a strict inequality \(<\)?
3. What if the expression was \( |3x - 1| \leq 7 \)? How would we approach it?
4. How can inequalities with multiple absolute values be solved?
5. How would the graph change for a quadratic inequality?

**Tip**: Always remember to check whether the inequality is strict ( \(<\) or \(>\) ) or includes equality ( \(\leq\) or \(\geq\) ) as it changes how you represent the solution on a number line.

Solve the inequality: |2x| ≤ 6 and tell me what that looks like on a number line.

Learn how to solve the absolute value inequality |2x| ≤ 6 and visualize the solution on a number line. This detailed step-by-step explanation covers breaking down the inequality and representing the solution as a closed interval on a number line. Suitable for students in Grades 7-9.

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