Let `y^(2) = ((x+5)(x-3))/(x-1)`, then all the real values of x, for which y has non-zero real values , are :

Options

A. `-5 lt x lt 1`
B. `x gt 15`
C. `-5 lt x lt 1` or `x gt 3`
D. None of these

Answer

To find the real values of \( x \) for which \( y \) has non-zero real values in the equation \( y^2 = \frac{(x+5)(x-3)}{(x-1)} \), we will analyze the conditions under which the right-hand side is positive.

### Step-by-Step Solution:

1. **Understanding the Equation**:
\[ y^2 = \frac{(x+5)(x-3)}{(x-1)} \]
Since \( y^2 \) must be non-negative and non-zero, the right-hand side must be positive.

2. **Case 1: Both Numerator and Denominator Positive**:
- The numerator \( (x+5)(x-3) > 0 \)
- The denominator \( (x-1) > 0 \)

**Finding the intervals for the numerator**:
- The critical points are \( x = -5 \) and \( x = 3 \).
- Testing intervals:
- For \( x < -5 \): Both factors are negative, product is positive.
- For \( -5 < x < 3 \): One factor is negative, product is negative.
- For \( x > 3 \): Both factors are positive, product is positive.
- Thus, \( (x+5)(x-3) > 0 \) for:
\[ x < -5 \quad \text{or} \quad x > 3 \]

**Finding the intervals for the denominator**:
- \( x - 1 > 0 \) gives \( x > 1 \).

**Combining the conditions**:
- From the numerator: \( (-\infty, -5) \cup (3, \infty) \)
- From the denominator: \( (1, \infty) \)
- The intersection gives:
\[ (-\infty, -5) \cup (3, \infty) \]

3. **Case 2: Both Numerator and Denominator Negative**:
- The numerator \( (x+5)(x-3) < 0 \)
- The denominator \( (x-1) < 0 \)

**Finding the intervals for the numerator**:
- As established before, \( (x+5)(x-3) < 0 \) for:
\[ -5 < x < 3 \]

**Finding the intervals for the denominator**:
- \( x - 1 < 0 \) gives \( x < 1 \).

**Combining the conditions**:
- From the numerator: \( (-5, 3) \)
- From the denominator: \( (-\infty, 1) \)
- The intersection gives:
\[ (-5, 1) \]

4. **Final Solution**:
- Combining both cases, we have:
\[ (-\infty, -5) \cup (-5, 1) \cup (3, \infty) \]
- This simplifies to:
\[ (-\infty, 1) \cup (3, \infty) \]

### Conclusion:
The values of \( x \) for which \( y \) has non-zero real values are:
\[ x \in (-\infty, 1) \cup (3, \infty) \]