Sure, let's solve the inequality step-by-step:
Given inequality:
[tex]\[
\left|18 + \frac{x}{2} \right| \geq 10
\][/tex]
To solve this absolute value inequality, we'll break it into two separate inequalities based on the definition of absolute value [tex]\( |A| \geq B \)[/tex] which translates to [tex]\( A \geq B \)[/tex] or [tex]\( A \leq -B \)[/tex].
#### Step 1: Split the inequality into two cases
1. Case 1:
[tex]\[
18 + \frac{x}{2} \geq 10
\][/tex]
2. Case 2:
[tex]\[
18 + \frac{x}{2} \leq -10
\][/tex]
#### Step 2: Solve each case individually
Case 1:
[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]
Case 2:
[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]
#### Step 3: Combine the solutions
The solutions from both cases are:
[tex]\[
x \leq -56 \quad \text{or} \quad x \geq -16
\][/tex]
So, the final solution to the inequality [tex]\(\left|18 + \frac{x}{2}\right| \geq 10\)[/tex] is:
[tex]\[
x \leq -56 \quad \text{or} \quad x \geq -16
\][/tex]
We can write this in interval notation as:
[tex]\[
(-\infty, -56] \cup [-16, \infty)
\][/tex]
Thus, the answer is:
[tex]\[
x \leq -56 \quad \text{or} \quad x \geq -16.
\][/tex]
