`y le 3x+1`
` x-ygt1` ,brgt Which ordered pair is in the set of all ordered pairs that satisfy the above system of inequalities?

Options

A. `(-1,-2)`
B. `(2,-1)`
C. `(1,2)`
D. `(-1,2)`

Answer

To solve the system of inequalities given by:

1. \( y \leq 3x + 1 \)
2. \( x - y > 1 \)

we need to check which of the provided ordered pairs satisfies both inequalities.

### Step 1: Check each option

We have four options to check:

1. Option 1: \( (-1, -2) \)
2. Option 2: \( (2, -1) \)
3. Option 3: \( (1, 2) \)
4. Option 4: \( (-1, 2) \)

#### Check Option 1: \( (-1, -2) \)

- Substitute \( x = -1 \) and \( y = -2 \) into the first inequality:

\[ -2 \leq 3(-1) + 1 \]
\[ -2 \leq -3 + 1 \]
\[ -2 \leq -2 \quad \text{(True)} \]

- Now check the second inequality:

\[ -1 - (-2) > 1 \]
\[ -1 + 2 > 1 \]
\[ 1 > 1 \quad \text{(False)} \]

**Conclusion**: Option 1 does not satisfy both inequalities.

---

#### Check Option 2: \( (2, -1) \)

- Substitute \( x = 2 \) and \( y = -1 \) into the first inequality:

\[ -1 \leq 3(2) + 1 \]
\[ -1 \leq 6 + 1 \]
\[ -1 \leq 7 \quad \text{(True)} \]

- Now check the second inequality:

\[ 2 - (-1) > 1 \]
\[ 2 + 1 > 1 \]
\[ 3 > 1 \quad \text{(True)} \]

**Conclusion**: Option 2 satisfies both inequalities.

---

#### Check Option 3: \( (1, 2) \)

- Substitute \( x = 1 \) and \( y = 2 \) into the first inequality:

\[ 2 \leq 3(1) + 1 \]
\[ 2 \leq 3 + 1 \]
\[ 2 \leq 4 \quad \text{(True)} \]

- Now check the second inequality:

\[ 1 - 2 > 1 \]
\[ -1 > 1 \quad \text{(False)} \]

**Conclusion**: Option 3 does not satisfy both inequalities.

---

#### Check Option 4: \( (-1, 2) \)

- Substitute \( x = -1 \) and \( y = 2 \) into the first inequality:

\[ 2 \leq 3(-1) + 1 \]
\[ 2 \leq -3 + 1 \]
\[ 2 \leq -2 \quad \text{(False)} \]

**Conclusion**: Option 4 does not satisfy both inequalities.

---

### Final Conclusion

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