The given absolute value inequality is:

$|x - 5| \geq 3$

To solve this, we need to break it into two separate inequalities based on the property of absolute values:

1. $x - 5 \geq 3$
2. $x - 5 \leq -3$

Step 1: Solve $x - 5 \geq 3$

$x - 5 \geq 3 \implies x \geq 3 + 5 \implies x \geq 8$

Step 2: Solve $x - 5 \leq -3$

$x - 5 \leq -3 \implies x \leq -3 + 5 \implies x \leq 2$

Solution:

Thus, the solution to the inequality is: $x \geq 8 \quad \text{or} \quad x \leq 2$

Graphically:

• For $x \geq 8$, the solution is all values greater than or equal to 8.
• For $x \leq 2$, the solution is all values less than or equal to 2.

This corresponds to the second number line in the image, where:

• There is an open circle at $x = 2$ and an arrow pointing to the left for values $x \leq 2$.
• There is an open circle at $x = 8$ and an arrow pointing to the right for values $x \geq 8$.

Thus, the correct answer is the second number line.

Do you want any more details on solving this?

Here are some related questions to expand on this:

1. How would the inequality change if the absolute value was less than instead of greater than?
2. How do open and closed circles differ when graphing inequalities?
3. What if the inequality was $|x - 5| > 3$? How would the solution change?
4. What are some practical applications of absolute value inequalities?
5. How can you graph inequalities involving quadratic functions?

Tip: When solving absolute value inequalities, remember to split them into two cases—one for the positive scenario and one for the negative scenario.