https://file.aiquickdraw.com/mathbot/file/1728873014562w9u4j2we.jpg

Let's solve the inequality given in the image step by step: 

### Inequality:  
\[
|x + 5| < 2
\]

### Step 1: Break the absolute value inequality into two cases
We need to solve two inequalities because the absolute value \(|x + 5|\) measures the distance from 0. For an inequality like \(|A| < B\), we break it into:

\[
-2 < x + 5 < 2
\]

### Step 2: Solve both inequalities
#### Left side:
\[
-2 < x + 5
\]
Subtract 5 from both sides:
\[
-7 < x \quad \Rightarrow \quad x > -7
\]

#### Right side:
\[
x + 5 < 2
\]
Subtract 5 from both sides:
\[
x < -3
\]

### Step 3: Combine the two inequalities
We now have:
\[
-7 < x < -3
\]

### Step 4: Write the solution in **interval notation**
The solution set is all \(x\)-values between \(-7\) and \(-3\), but not including the endpoints (since the inequality is strict). Therefore, the interval notation is:

\[
(-7, -3)
\]

### Step 5: Identify the correct graph
Looking at the graphs provided:
- We need an open interval from \(-7\) to \(-3\), which excludes both \(-7\) and \(-3\). 
- The correct graph is **option E**, which highlights the interval \((-7, -3)\) with open circles at both ends.

### Final Answer
- **Interval notation**: \((-7, -3)\)  
- **Correct graph**: **Option E**

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Do you have any questions or want further details? 

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Here are five related questions:
1. What changes if the inequality was \(|x + 5| \leq 2\)?
2. How do we solve absolute value inequalities where the inequality is \(\geq\) or \(>\)?
3. How would you graph the solution for \(|x - 1| < 3\)?
4. What happens if the absolute value inequality has no solutions (e.g., \(|x + 5| < -2\))?
5. How do you solve compound inequalities involving \(\text{and/or}\) conditions?

**Tip:** When solving absolute value inequalities, always check whether the interval needs open or closed endpoints based on the inequality sign.

Learn how to solve the absolute value inequality |x + 5| < 2 with a step-by-step breakdown. This problem explores key algebraic concepts, including absolute value and inequalities, suitable for Grades 8-10 students.

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