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Let's solve this problem by checking which of the given ordered pairs satisfies both inequalities.

The system of inequalities is:
1. \( y \leq 3x + 1 \)
2. \( x - y > 1 \)

Let's evaluate each given ordered pair to determine which one satisfies both inequalities.

### Option A: \((-1, -2)\)
- For \( y \leq 3x + 1 \):
  \[
  -2 \leq 3(-1) + 1 \implies -2 \leq -3 + 1 \implies -2 \leq -2 \quad \text{(True)}
  \]
- For \( x - y > 1 \):
  \[
  -1 - (-2) > 1 \implies -1 + 2 > 1 \implies 1 > 1 \quad \text{(False)}
  \]

This pair does not satisfy both inequalities.

### Option B: \((2, -1)\)
- For \( y \leq 3x + 1 \):
  \[
  -1 \leq 3(2) + 1 \implies -1 \leq 6 + 1 \implies -1 \leq 7 \quad \text{(True)}
  \]
- For \( x - y > 1 \):
  \[
  2 - (-1) > 1 \implies 2 + 1 > 1 \implies 3 > 1 \quad \text{(True)}
  \]

This pair satisfies both inequalities.

### Option C: \((1, 2)\)
- For \( y \leq 3x + 1 \):
  \[
  2 \leq 3(1) + 1 \implies 2 \leq 3 + 1 \implies 2 \leq 4 \quad \text{(True)}
  \]
- For \( x - y > 1 \):
  \[
  1 - 2 > 1 \implies -1 > 1 \quad \text{(False)}
  \]

This pair does not satisfy both inequalities.

### Option D: \((-1, 2)\)
- For \( y \leq 3x + 1 \):
  \[
  2 \leq 3(-1) + 1 \implies 2 \leq -3 + 1 \implies 2 \leq -2 \quad \text{(False)}
  \]

This pair does not satisfy the first inequality.

### Conclusion
The ordered pair that satisfies both inequalities is **Option B: \((2, -1)\)**.

Would you like more details or have any questions about this solution?

Here are some related questions to deepen your understanding:

1. How do we graph the system of inequalities on a coordinate plane?
2. What is the significance of testing each inequality with ordered pairs?
3. How can we determine the solution set for a system of linear inequalities?
4. Can you solve another example where both inequalities have a different slope?
5. What happens if both inequalities are strict (using < or >)?

**Tip:** When solving inequalities involving ordered pairs, always verify if each pair satisfies **both** inequalities to be part of the solution set.

Which ordered pair is in the solution set of the system of inequalities y ≤ 3x + 1 and x − y > 1?

Learn how to solve a system of inequalities involving y ≤ 3x + 1 and x - y > 1. This example walks through testing ordered pairs to find which satisfies both inequalities. Suitable for Grades 8-10.

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