In each of the following questions a pair of equations is given. You have to find out the value of x and y and give answer -----
(A)If x `lt` y (B) If x `le`y
( C ) If x = y ( D ) If x `gt` y
(E ) If x `ge` y
(1) `x^2` + 9x + 14 = 0
(2) `y^2` + y - 2 = 0

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation for \( x \)
The first equation is:
\[ x^2 + 9x + 14 = 0 \]
To factor this quadratic equation, we look for two numbers that multiply to \( 14 \) and add to \( 9 \). The numbers \( 7 \) and \( 2 \) satisfy this condition. Thus, we can factor the equation as:
\[ (x + 7)(x + 2) = 0 \]
Setting each factor to zero gives us:
\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
So, the possible values for \( x \) are \( -7 \) and \( -2 \).

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ y^2 + y - 2 = 0 \]
To factor this quadratic equation, we look for two numbers that multiply to \( -2 \) and add to \( 1 \). The numbers \( 2 \) and \( -1 \) satisfy this condition. Thus, we can factor the equation as:
\[ (y + 2)(y - 1) = 0 \]
Setting each factor to zero gives us:
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]
So, the possible values for \( y \) are \( -2 \) and \( 1 \).

### Step 3: Compare the values of \( x \) and \( y \)
Now we have the following pairs of values:
- For \( x \): \( -7 \) and \( -2 \)
- For \( y \): \( -2 \) and \( 1 \)

Now we will compare each value of \( x \) with each value of \( y \):

1. **When \( x = -7 \)**:
- Compare with \( y = -2 \): \( -7 < -2 \) (True)
- Compare with \( y = 1 \): \( -7 < 1 \) (True)

2. **When \( x = -2 \)**:
- Compare with \( y = -2 \): \( -2 = -2 \) (True)
- Compare with \( y = 1 \): \( -2 < 1 \) (True)

### Conclusion
From the comparisons, we can see:
- \( x = -7 \) is less than both values of \( y \).
- \( x = -2 \) is equal to \( y = -2 \) and less than \( y = 1 \).

Thus, the relationship that holds true for all cases is:
\[ x \leq y \]

### Final Answer
The answer is (B) If \( x \leq y \).

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