https://file.aiquickdraw.com/mathbot/file/1727103848241cv6mmvkd.jpg

The given inequality is:

\[
|r - 60| > 20
\]

To solve this absolute value inequality, we break it into two cases based on the definition of absolute values.

### Case 1: Positive Case
\[
r - 60 > 20
\]
Add 60 to both sides:
\[
r > 80
\]

### Case 2: Negative Case
\[
-(r - 60) > 20
\]
This simplifies to:
\[
r - 60 < -20
\]
Add 60 to both sides:
\[
r < 40
\]

### Solution:
The combined solution is:
\[
r < 40 \quad \text{or} \quad r > 80
\]

### Graphing the Solution:
On a number line, we will represent this as two separate rays:
- One ray extending from \(r = 40\) to the left (indicating \(r < 40\)).
- Another ray extending from \(r = 80\) to the right (indicating \(r > 80\)).

Since the inequality is strict (no equals sign), we use open circles at \(r = 40\) and \(r = 80\) to indicate that these points are not included in the solution.

Let me know if you'd like further details or help with graphing this, and feel free to ask any questions!

### Follow-up Questions:
1. What happens if the inequality is \( |r - 60| \geq 20 \)?
2. How do absolute value inequalities differ from linear inequalities?
3. What is the general process for solving inequalities involving absolute values?
4. Can you explain how to graph rays and line segments on a number line?
5. How would the solution change if the inequality was \( |r - 60| < 20 \)?

### Tip:
When graphing inequalities, always pay attention to whether the inequality is strict (like \(>\) or \(<\)) or inclusive (like \( \geq \) or \( \leq \)), as this determines whether to use open or closed circles on the number line.

Solve for r and graph the solution: |r - 60| > 20

Learn how to solve the absolute value inequality |r - 60| > 20. This problem covers key algebraic concepts, including solving absolute value inequalities, breaking them into two cases, and graphing the solution. Suitable for students in Grades 8-10.

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