` x^(2) - 10x+24 = 0, y^(2) - 14y+48 = 0`

Options

A. if ` x gt y`
B. if ` x lt y`
C. if ` x ge y`
D. if ` x le y`

Explanation

To solve the equations \( x^2 - 10x + 24 = 0 \) and \( y^2 - 14y + 48 = 0 \), we will factor each quadratic equation step-by-step.

### Step 1: Solve the first equation \( x^2 - 10x + 24 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -10 \), and \( c = 24 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( 24 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)).
- The numbers are \( -4 \) and \( -6 \) because:
\[ -4 \times -6 = 24 \quad \text{and} \quad -4 + -6 = -10 \]
3. **Write the factored form**:
\[ (x - 4)(x - 6) = 0 \]
4. **Set each factor to zero**:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
5. **Solutions for \( x \)**: The possible values for \( x \) are \( 4 \) and \( 6 \).

### Step 2: Solve the second equation \( y^2 - 14y + 48 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = -14 \), and \( c = 48 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( 48 \) and add up to \( -14 \).
- The numbers are \( -6 \) and \( -8 \) because:
\[ -6 \times -8 = 48 \quad \text{and} \quad -6 + -8 = -14 \]
3. **Write the factored form**:
\[ (y - 6)(y - 8) = 0 \]
4. **Set each factor to zero**:
\[ y - 6 = 0 \quad \Rightarrow \quad y = 6 \]
\[ y - 8 = 0 \quad \Rightarrow \quad y = 8 \]
5. **Solutions for \( y \)**: The possible values for \( y \) are \( 6 \) and \( 8 \).

### Step 3: Determine the relationship between \( x \) and \( y \)

- The values we found are:
- \( x = 4 \) or \( x = 6 \)
- \( y = 6 \) or \( y = 8 \)

- Now we can analyze the relationships:
- If \( x = 4 \), then \( y = 6 \) (so \( x < y \)).
- If \( x = 6 \), then \( y = 6 \) (so \( x = y \)).
- If \( y = 8 \), then \( x = 6 \) (so \( x < y \)).

### Conclusion

The relationship can be summarized as:
- \( x \leq y \)

Thus, the final answer is that \( x \) is less than or equal to \( y \).

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