The ordered pairs (3,-1) satisfies which of the following inequalities?
I. `x+3y gt0`
II. `2x+3y gt2`
III. `x+y lt 0`

Options

A. I only
B. II only
C. I and III only
D. II and III only

Answer

To determine which inequalities the ordered pair (3, -1) satisfies, we will evaluate each inequality one by one.

### Step 1: Evaluate Inequality I: \( x + 3y > 0 \)

1. Substitute \( x = 3 \) and \( y = -1 \) into the inequality:
\[ 3 + 3(-1) > 0 \]
2. Simplify the expression:
\[ 3 - 3 > 0 \]
\[ 0 > 0 \]
3. This statement is false since \( 0 \) is not greater than \( 0 \).

### Step 2: Evaluate Inequality II: \( 2x + 3y > 2 \)

1. Substitute \( x = 3 \) and \( y = -1 \) into the inequality:
\[ 2(3) + 3(-1) > 2 \]
2. Simplify the expression:
\[ 6 - 3 > 2 \]
\[ 3 > 2 \]
3. This statement is true.

### Step 3: Evaluate Inequality III: \( x + y < 0 \)

1. Substitute \( x = 3 \) and \( y = -1 \) into the inequality:
\[ 3 + (-1) < 0 \]
2. Simplify the expression:
\[ 3 - 1 < 0 \]
\[ 2 < 0 \]
3. This statement is false since \( 2 \) is not less than \( 0 \).

### Conclusion

The ordered pair (3, -1) satisfies only the second inequality \( 2x + 3y > 2 \).

### Summary of Results:
- Inequality I: False
- Inequality II: True
- Inequality III: False

Thus, the answer is that the ordered pair (3, -1) satisfies only **Inequality II**.

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