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.