Let's begin by solving the absolute value inequality:

$|x - 2| \geq 3$

Step 1: Break down the absolute value inequality into two cases.

1. Case 1: $x - 2 \geq 3$ $x \geq 5$

2. Case 2: $x - 2 \leq -3$ $x \leq -1$

So, the solution to the inequality $|x - 2| \geq 3$ is $x \geq 5$ or $x \leq -1$.

Step 2: Logical operator for solutions

Since the absolute value represents a distance, we are considering the numbers whose distance from 2 is greater than or equal to 3. The word "or" is key here because we are looking for values either on the left or right of the critical points, not in between.

So, the correct choice in part A would be: 2. "We should be thinking that solutions are any $x$-values that are solutions for either of our inequalities, i.e., we should be thinking 'OR'."

Step 3: Graphing the solution

Now, let's examine part B. The solution $x \geq 5$ or $x \leq -1$ will be a graph with shaded regions extending left from $-1$ and right from $5$, including $-1$ and $5$ as the boundary points. Looking at the graph options:

• Option 1 shows $x \leq -1$ and $x \geq 5$, with filled circles at $-1$ and 5, indicating that these values are included in the solution. This is the correct graph.

Thus, for part B, the correct answer is 1.

Let me know if you need further clarification on any of the steps.

Follow-up questions:

1. What happens if the inequality had been $|x - 2| < 3$?
2. How does the solution change if the inequality sign is $\leq$ instead of $\geq$?
3. What is the reasoning behind splitting the absolute value into two inequalities?
4. How would the graph change if the inequality was strict (e.g., $>$ or $<$)?
5. Can you explain why the word "OR" is used instead of "AND" in this situation?

Tip: When solving absolute value inequalities, always remember that $|A| \geq B$ translates to $A \geq B$ or $A \leq -B$.