https://file.aiquickdraw.com/mathbot/file/17265508865107um8peka.jpg

The inequality shown in the image is:

\[
\left| 18 + \frac{x}{2} \right| \geq 10
\]

To solve this absolute value inequality, we break it into two cases:

### Case 1: Positive Case
\[
18 + \frac{x}{2} \geq 10
\]
Subtracting 18 from both sides:
\[
\frac{x}{2} \geq 10 - 18
\]
\[
\frac{x}{2} \geq -8
\]
Multiplying both sides by 2:
\[
x \geq -16
\]

### Case 2: Negative Case
\[
-(18 + \frac{x}{2}) \geq 10
\]
This simplifies to:
\[
-18 - \frac{x}{2} \geq 10
\]
Adding 18 to both sides:
\[
-\frac{x}{2} \geq 28
\]
Multiplying both sides by -2 (and reversing the inequality):
\[
x \leq -56
\]

### Final Solution
The solution to the inequality is:
\[
x \geq -16 \quad \text{or} \quad x \leq -56
\]

This can be written as the union of two intervals:
\[
x \in (-\infty, -56] \cup [-16, \infty)
\]

Let me know if you need further clarification or details.

Here are 5 related questions for practice:
1. How would you solve the inequality \( |x - 5| \geq 3 \)?
2. Can you solve the inequality \( |2x + 7| \leq 9 \)?
3. How do you interpret the solution to an inequality with absolute values geometrically?
4. What happens if the absolute value inequality involves \( \leq \) instead of \( \geq \)?
5. How would the solution change if the inequality was \( |18 + \frac{x}{2}| > 10 \)?

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

Learn how to solve the absolute value inequality |18 + x/2| ≥ 10 with step-by-step breakdowns for both positive and negative cases. This problem highlights key concepts of algebra and inequalities, making it suitable for students in Grades 9-10.

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