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The question asks you to rewrite the absolute value inequality \( |9x + 6| < 24 \) as two linear inequalities. Let’s walk through the solution step by step:

### Step 1: Understand the Absolute Value Inequality
The inequality \( |9x + 6| < 24 \) means that the expression inside the absolute value must lie between \(-24\) and \(24\). This translates to:

\[
-24 < 9x + 6 < 24
\]

### Step 2: Break it into Two Inequalities
We can rewrite the inequality above as two separate inequalities:

1. \(9x + 6 < 24\)  
2. \(9x + 6 > -24\) (or equivalently, \(-24 < 9x + 6\))

### Step 3: Solve Both Inequalities
Now, solve each inequality for \(x\).

1. **For \(9x + 6 < 24\)**:
   \[
   9x < 24 - 6
   \]
   \[
   9x < 18
   \]
   \[
   x < 2
   \]

2. **For \(-24 < 9x + 6\)**:
   \[
   -24 - 6 < 9x
   \]
   \[
   -30 < 9x
   \]
   \[
   \frac{-30}{9} < x \quad \text{or} \quad x > -\frac{10}{3}
   \]

### Step 4: Write the Final Solution
The two linear inequalities are:

\[
-\frac{10}{3} < x < 2
\]

Since both inequalities must hold together, the answer involves **"and"** (both conditions need to be satisfied).

### Final Answer:
- Inequalities:  
  \(-\frac{10}{3} < x\) and \(x < 2\)
- Join with: **and**

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Do you want further details or have any questions?  
Here are 5 related questions:
1. What changes if the inequality was \( |9x + 6| \leq 24 \)?
2. How do you solve absolute value inequalities with \(\geq\) or \(>\) instead of \(<\)?
3. How do you graph solutions of these types of inequalities on a number line?
4. What happens if the absolute value inequality has no solution?
5. How do you handle absolute value inequalities involving \( \neq \) or multiple variables?

**Tip:** When solving absolute value inequalities, always consider both the positive and negative cases!

Rewrite the absolute value inequality |9x + 6| < 24 as two linear inequalities.

Learn how to rewrite and solve the absolute value inequality |9x + 6| < 24 by breaking it into two linear inequalities. This detailed step-by-step solution involves key algebraic concepts such as inequalities and absolute value properties. Suitable for students in Grades 9-11.

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