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.