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The inequality provided is \( |x| \geq 3 \).

To solve this inequality and mark the solutions on the number line, let's break it down step by step.

### Step 1: Understanding the Absolute Value Inequality
The inequality \( |x| \geq 3 \) means that the distance of \( x \) from 0 on the number line is greater than or equal to 3. This gives us two possible cases:

1. \( x \geq 3 \)
2. \( x \leq -3 \)

### Step 2: Graphing the Solutions
To represent the solution on the number line:

- The solution \( x \geq 3 \) is represented by a ray starting from 3 and extending to the right (positive infinity). Since the inequality is "greater than or equal to," we include the point 3 by using a filled (closed) circle at 3.
- The solution \( x \leq -3 \) is represented by a ray starting from -3 and extending to the left (negative infinity). Similarly, we include the point -3 by using a filled (closed) circle at -3.

### Step 3: Plotting the Number Line
The number line will look like this:

```
<---●========●--->
   -3        3
```

Here:
- The filled circles (●) at -3 and 3 indicate that these points are included in the solution set.
- The arrows (---) indicate that the solution continues indefinitely in both directions (less than -3 and greater than 3).

### Conclusion
The solution set for \( |x| \geq 3 \) includes all numbers less than or equal to -3 and all numbers greater than or equal to 3.

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How would the solution change if the inequality was \( |x| > 3 \)?
2. How can you express the solution in interval notation?
3. What does \( |x| \leq 3 \) represent on a number line?
4. Can you solve and graph \( |x - 2| \geq 4 \) on a number line?
5. What is the difference between strict and non-strict inequalities in absolute value equations?

**Tip:** When dealing with absolute value inequalities, always consider both the positive and negative scenarios to capture the complete solution set.

Learn how to mark the solutions of the absolute value inequality |x| >= 3 on the number line with a step-by-step solution. This problem covers absolute value inequalities and their representation on a number line. Suitable for students in Grades 7-9.

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