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Let's solve the given compound inequality:

We are provided with two inequalities:
1. \( 24 > 5x + 9 \)
2. \( 5x + 9 \geq 69 \)

### Solving Inequality 1:
\[
24 > 5x + 9
\]
- Subtract 9 from both sides:
\[
24 - 9 > 5x
\]
\[
15 > 5x
\]
- Divide by 5:
\[
3 > x \quad \text{or} \quad x < 3
\]

### Solving Inequality 2:
\[
5x + 9 \geq 69
\]
- Subtract 9 from both sides:
\[
5x \geq 60
\]
- Divide by 5:
\[
x \geq 12
\]

### Final Solution:
We now have two conditions:
- \( x < 3 \)
- \( x \geq 12 \)

This represents a "disjoint" or "or" condition. Therefore, the solution is \( x < 3 \) **or** \( x \geq 12 \).

### Inequality Notation:
In interval notation, this would be:
\[
(-\infty, 3) \cup [12, \infty)
\]

For the graph on the number line:
- Shade the region to the left of \( x = 3 \) (open circle at 3).
- Shade the region starting from \( x = 12 \) and going to the right (closed circle at 12).

Does that explanation help? Feel free to ask for more details if needed!

Here are 5 related questions you can explore:
1. What are the properties of compound inequalities?
2. How do we graph compound inequalities with disjoint intervals?
3. What happens when the inequality involves a "less than or equal to" sign?
4. Can compound inequalities have no solutions?
5. How would the solution change if both inequalities were connected with "and" instead of "or"?

**Tip:** Always double-check whether the compound inequality is connected with "and" or "or" before solving!

Solve for x and graph the solution on the number line below: 24 > 5x + 9 or 5x + 9 ≥ 69

Learn how to solve the compound inequality 24 > 5x + 9 or 5x + 9 ≥ 69. This problem involves algebraic manipulation, including solving inequalities and graphing the solution on a number line. Suitable for students in Grades 9-10.

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