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The image shows a problem where several cards need to be sorted into four groups based on inequalities and their corresponding solutions.

Here are the cards:
1. **Inequalities/Equations:**
   - \( |x - 2| \leq 1 \)
   - \( |x + 1| \leq 2 \)
   - \( |x - 1| \geq 2 \)

2. **Statements about solutions:**
   - "The set of all numbers at least 2 units away from 1."
   - "The set of all numbers at most 1 unit away from 2."
   - "The set of all numbers at least 1 unit away from −2."
   - "x = −2 is a solution. x = 0 is a solution. x = 3 is not a solution."
   - "x = −2 is not a solution. x = 0 is not a solution. x = 3 is a solution."

3. **Graphs of solution sets:**
   - A graph where the shaded region is from \( -1 \) to \( 3 \), excluding 3.
   - A graph where the shaded region is from \( -1 \) to \( 1 \).

The goal is to group the cards based on their relationships.

### Sorting Strategy:

1. **Group for \( |x - 2| \leq 1 \):**
   - This inequality represents all numbers that are within 1 unit of 2. Therefore, the statement "The set of all numbers at most 1 unit away from 2" fits here. Additionally, the graph showing the interval from 1 to 3 fits because these values are 1 unit away from 2.
   - The solution set is \( 1 \leq x \leq 3 \), so the card with "x = −2 is not a solution, x = 0 is not a solution, x = 3 is a solution" fits.

2. **Group for \( |x + 1| \leq 2 \):**
   - This represents the set of all numbers that are at most 2 units away from −1. The solution set is \( -3 \leq x \leq 1 \). So, the graph showing the interval from \( -3 \) to \( 1 \) fits, as well as the card "x = −2 is a solution, x = 0 is a solution, x = 3 is not a solution."

3. **Group for \( |x - 1| \geq 2 \):**
   - This represents the set of all numbers that are at least 2 units away from 1. The card "The set of all numbers at least 2 units away from 1" fits here, and the solution is \( x \leq -1 \) or \( x \geq 3 \), so a graph that reflects this set also fits here.

4. **Remaining cards:**
   - The remaining cards seem to describe specific sets that could align with the intervals given in the inequalities or solution sets.

Would you like a detailed breakdown of each inequality or a specific explanation of any of these groups?

### Related Questions:
1. How do we interpret absolute value inequalities graphically?
2. What are the steps to solving \( |x - a| \leq b \)?
3. How can we verify solutions of absolute value inequalities using substitution?
4. How do solution sets differ between \( |x - a| \leq b \) and \( |x - a| \geq b \)?
5. What are the common mistakes students make when solving absolute value inequalities?

**Tip:** When solving absolute value inequalities, break them into two cases (positive and negative) to simplify the solution process.

Sort these cards into four groups based on the inequalities and corresponding solutions.

In this problem, students are tasked with sorting cards based on absolute value inequalities and their corresponding graphical solutions. It includes key algebraic concepts such as solving inequalities, graphing solutions, and analyzing solution sets. Suitable for students in Grades 8-10.

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