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The given inequality is an absolute value inequality:

\[
|5x + 7| > 24
\]

To rewrite this absolute value inequality as two linear inequalities, you use the following property of absolute value inequalities:

- If \( |A| > B \), then \( A > B \) or \( A < -B \).

In this case, \( A = 5x + 7 \) and \( B = 24 \). So, we have:

1. \( 5x + 7 > 24 \)
2. \( 5x + 7 < -24 \)

Now, let's solve each inequality:

### First inequality:
\[
5x + 7 > 24
\]
Subtract 7 from both sides:
\[
5x > 17
\]
Divide by 5:
\[
x > \frac{17}{5}
\]

### Second inequality:
\[
5x + 7 < -24
\]
Subtract 7 from both sides:
\[
5x < -31
\]
Divide by 5:
\[
x < \frac{-31}{5}
\]

So, the two inequalities are:
\[
x > \frac{17}{5} \quad \text{or} \quad x < \frac{-31}{5}
\]

Therefore, the correct answer is:
- First inequality: \( x > \frac{17}{5} \)
- Second inequality: \( x < \frac{-31}{5} \)
- Use "or" to join the inequalities.

Would you like more details or have any other questions?

---

Here are five questions to deepen your understanding:
1. How would the solution change if the inequality were \( |5x + 7| \leq 24 \)?
2. Can you solve a similar inequality with \( |2x - 3| > 15 \)?
3. How do you handle absolute value inequalities when solving for multiple variables?
4. What happens if we have an equality instead of inequality, like \( |5x + 7| = 24 \)?
5. Why do we use "or" instead of "and" for this inequality?

**Tip**: For absolute value inequalities of the form \( |A| > B \), always remember to split them into two cases: \( A > B \) or \( A < -B \).

Consider the following absolute value inequality: |5x + 7| > 24. Rewrite the given inequality as two linear inequalities.

Learn how to solve the absolute value inequality |5x + 7| > 24 by rewriting it as two linear inequalities. This problem covers key algebraic concepts and absolute value properties. Suitable for high school students in Grades 9-11.

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