If `A={x in R: x^(2)+6x-7 lt 0} and B={ x in R: x^(2)+9x+14 gt 0}`, then which of the following is/are correct ?
1. `A cap B = {x in R: -2 x lt x lt 1}`
2. `AUB = {x in R: -7 lt x lt -2}`
Select the correct answer using the codes given below :

Options

A. A) Only 1
B. B) Only 2
C. C) Both 1 and 2
D. D) Neither 1 nor 2

Answer

To solve the problem, we need to analyze the sets \( A \) and \( B \) based on the given inequalities.

### Step 1: Determine the set \( A \)

The set \( A \) is defined as:
\[ A = \{ x \in \mathbb{R} : x^2 + 6x - 7 < 0 \} \]

First, we need to factor the quadratic expression \( x^2 + 6x - 7 \).

1. **Finding the roots:**
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 6, c = -7 \).
\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} = \frac{-6 \pm \sqrt{36 + 28}}{2} = \frac{-6 \pm \sqrt{64}}{2} = \frac{-6 \pm 8}{2} \]
This gives us the roots:
\[ x = 1 \quad \text{and} \quad x = -7 \]

2. **Sign analysis:**
We analyze the sign of \( x^2 + 6x - 7 \) in the intervals determined by the roots \( -7 \) and \( 1 \):
- For \( x < -7 \): Both factors \( (x + 7) \) and \( (x - 1) \) are negative, so the product is positive.
- For \( -7 < x < 1 \): \( (x + 7) \) is positive and \( (x - 1) \) is negative, so the product is negative.
- For \( x > 1 \): Both factors are positive, so the product is positive.

Thus, the solution to \( x^2 + 6x - 7 < 0 \) is:
\[ A = (-7, 1) \]

### Step 2: Determine the set \( B \)

The set \( B \) is defined as:
\[ B = \{ x \in \mathbb{R} : x^2 + 9x + 14 > 0 \} \]

1. **Finding the roots:**
We factor the quadratic expression \( x^2 + 9x + 14 \):
\[ x^2 + 9x + 14 = (x + 7)(x + 2) \]
The roots are:
\[ x = -7 \quad \text{and} \quad x = -2 \]

2. **Sign analysis:**
We analyze the sign of \( (x + 7)(x + 2) > 0 \):
- For \( x < -7 \): Both factors are negative, so the product is positive.
- For \( -7 < x < -2 \): \( (x + 7) \) is positive and \( (x + 2) \) is negative, so the product is negative.
- For \( x > -2 \): Both factors are positive, so the product is positive.

Thus, the solution to \( x^2 + 9x + 14 > 0 \) is:
\[ B = (-\infty, -7) \cup (-2, \infty) \]

### Step 3: Find \( A \cap B \)

To find \( A \cap B \), we look for the intersection of the intervals:
- \( A = (-7, 1) \)
- \( B = (-\infty, -7) \cup (-2, \infty) \)

The intersection is:
\[ A \cap B = (-7, 1) \cap \left( (-\infty, -7) \cup (-2, \infty) \right) = (-2, 1) \]

### Step 4: Find \( A \cup B \)

To find \( A \cup B \), we combine the intervals:
\[ A \cup B = (-7, 1) \cup \left( (-\infty, -7) \cup (-2, \infty) \right) = (-\infty, -7) \cup (-2, 1) \]

### Conclusion

Now we can check the statements given in the question:

1. **Statement 1:** \( A \cap B = \{ x \in \mathbb{R} : -2 < x < 1 \} \) is **correct**.
2. **Statement 2:** \( A \cup B = \{ x \in \mathbb{R} : -7 < x < -2 \} \) is **incorrect**.

### Final Answer

The correct answer is **1 only**.