https://file.aiquickdraw.com/mathbot/file/1727306166793tjmohvjg.jpg

The inequality shown in the image is:

\[
|x + 2| > 9
\]

To solve this absolute value inequality, we break it into two cases based on the definition of absolute value:

1. \( x + 2 > 9 \)
2. \( x + 2 < -9 \)

### Case 1:
Solve \( x + 2 > 9 \):
\[
x > 9 - 2
\]
\[
x > 7
\]

### Case 2:
Solve \( x + 2 < -9 \):
\[
x < -9 - 2
\]
\[
x < -11
\]

Thus, the solution to the inequality \( |x + 2| > 9 \) is:
\[
x > 7 \quad \text{or} \quad x < -11
\]

This means \( x \) must be either greater than 7 or less than -11.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How would you solve \( |x - 5| > 12 \)?
2. What happens if the inequality is \( |x + 2| \leq 9 \) instead?
3. Can absolute value inequalities have no solution?
4. How can you graph the solution of \( |x + 2| > 9 \) on a number line?
5. How do you interpret the solution of an inequality involving absolute values in real-world contexts?

**Tip:** When solving absolute value inequalities, always remember to handle both the positive and negative cases separately.

Learn how to solve the absolute value inequality |x + 2| > 9 with step-by-step instructions. Explore how to break the inequality into two cases and solve for x > 7 or x < -11. Suitable for students in Grades 7-9.

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