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The inequality given is:

\[
|x - 3| \geq 5
\]

To solve this absolute value inequality, we break it into two separate inequalities based on the definition of absolute value.

### Step 1: Split into two cases
1. \( x - 3 \geq 5 \)
2. \( x - 3 \leq -5 \)

### Step 2: Solve each inequality

#### Case 1: \( x - 3 \geq 5 \)
Add 3 to both sides:
\[
x \geq 8
\]

#### Case 2: \( x - 3 \leq -5 \)
Add 3 to both sides:
\[
x \leq -2
\]

### Final solution:
The solution set for the inequality is:

\[
x \geq 8 \quad \text{or} \quad x \leq -2
\]

In interval notation, this is:

\[
(-\infty, -2] \cup [8, \infty)
\]

Would you like more details or have any questions about this process?

Here are 5 related questions to explore further:
1. How do you solve absolute value inequalities with "<" instead of "≥"?
2. What is the graphical representation of the solution to this inequality on a number line?
3. How does solving absolute value equations differ from inequalities?
4. Can you apply this method to inequalities involving quadratic expressions inside the absolute value?
5. How would the solution change if the inequality was strict, \( |x - 3| > 5 \)?

**Tip:** Always remember, for absolute value inequalities, use "or" when the inequality is "≥" or ">", and "and" when it is "≤" or "<".

Discover how to solve the absolute value inequality |x - 3| ≥ 5 with detailed steps. This solution breaks down the inequality into two separate cases and explains each step clearly, suitable for students in Grades 8-10.

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