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!