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To solve the inequality \( |x - 5| \geq 2 \) and graph it on the number line, follow these steps:

### 1. Break the absolute value inequality into two separate inequalities:
- For absolute value inequalities, we handle \( |x - 5| \geq 2 \) as two cases:
  \[
  x - 5 \geq 2 \quad \text{or} \quad x - 5 \leq -2
  \]

### 2. Solve each inequality:
- **First case:** \( x - 5 \geq 2 \)
  \[
  x \geq 7
  \]
  
- **Second case:** \( x - 5 \leq -2 \)
  \[
  x \leq 3
  \]

### 3. Combine the solution:
The solution to the inequality is:
\[
x \geq 7 \quad \text{or} \quad x \leq 3
\]

This represents two disjoint intervals:
- \( x \in (-\infty, 3] \)
- \( x \in [7, \infty) \)

### 4. Graph the solution on the number line:
- At \( x = 3 \), put a **closed dot** since \( x \leq 3 \) includes \( 3 \), and shade the number line to the left towards \( -\infty \).
- At \( x = 7 \), put another **closed dot** since \( x \geq 7 \) includes \( 7 \), and shade the number line to the right towards \( \infty \).

This graph shows two distinct intervals.

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Would you like more details about any of the steps, or do you have any questions?

### Related Questions:
1. How do you solve absolute value inequalities when the sign is \( < \) instead of \( \geq \)?
2. What is the general rule for solving \( |x - a| \geq b \)?
3. How can inequalities be represented using interval notation?
4. What is the difference between open and closed dots on a number line?
5. How does the solution of an absolute value inequality change if we have \( > \) instead of \( \geq \)?

### Tip:
When graphing inequalities, remember that a closed dot is used when the variable is included (i.e., \( \geq \) or \( \leq \)), while an open dot is used for strict inequalities (i.e., \( > \) or \( < \)).

Graph the solution to the inequality |x - 5| ≥ 2 on the number line.

Learn how to solve the absolute value inequality |x - 5| ≥ 2 by breaking it into two cases and graphing the solution on a number line. This solution covers key algebraic concepts, including absolute value properties, inequality solving techniques, and number line graphing. Suitable for students in Grades 7-9.

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