To solve the inequality
[tex]\[
\left|18 + \frac{x}{2}\right| \geq 10,
\][/tex]
we'll proceed step by step.
1. Understand the Absolute Value Inequality:
[tex]\[
\left| A \right| \geq B \implies A \leq -B \text{ or } A \geq B.
\][/tex]
Here, [tex]\( A = 18 + \frac{x}{2} \)[/tex] and [tex]\( B = 10 \)[/tex]. Thus, we can rewrite the inequality as:
[tex]\[
18 + \frac{x}{2} \leq -10 \quad \text{or} \quad 18 + \frac{x}{2} \geq 10.
\][/tex]
2. Solve Each Case Separately:
- Case 1: [tex]\( 18 + \frac{x}{2} \leq -10 \)[/tex]
[tex]\[
18 + \frac{x}{2} \leq -10
\][/tex]
Subtract 18 from both sides:
[tex]\[
\frac{x}{2} \leq -10 - 18
\][/tex]
[tex]\[
\frac{x}{2} \leq -28
\][/tex]
Multiply both sides by 2:
[tex]\[
x \leq -56
\][/tex]
- Case 2: [tex]\( 18 + \frac{x}{2} \geq 10 \)[/tex]
[tex]\[
18 + \frac{x}{2} \geq 10
\][/tex]
Subtract 18 from both sides:
[tex]\[
\frac{x}{2} \geq 10 - 18
\][/tex]
[tex]\[
\frac{x}{2} \geq -8
\][/tex]
Multiply both sides by 2:
[tex]\[
x \geq -16
\][/tex]
3. Combine the Results:
[tex]\[
x \leq -56 \quad \text{or} \quad x \geq -16.
\][/tex]
So, the solution to the inequality [tex]\(\left|18 + \frac{x}{2}\right| \geq 10\)[/tex] is:
[tex]\(\boxed{x \leq -56 \quad \text{or} \quad x \geq -16}\)[/tex].
