If x is not equal to 1 and `y = 1/(x - 1)`, then which one of the following cannot be the value of y?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the equation given and the options provided.

Given:
\[ y = \frac{1}{x - 1} \]
where \( x \neq 1 \).

We need to determine which of the following values cannot be the value of \( y \):
1. \( 0 \)
2. \( 1 \)
3. \( 2 \)
4. \( 3 \)

### Step 1: Analyze the equation
The equation \( y = \frac{1}{x - 1} \) indicates that \( y \) is the reciprocal of \( (x - 1) \).

### Step 2: Determine the conditions for \( y \)
Since \( y \) is defined as the reciprocal of \( (x - 1) \), we can see that:
- If \( x - 1 \) approaches \( 0 \) (i.e., \( x \) approaches \( 1 \)), \( y \) will approach \( \infty \) or \( -\infty \).
- If \( x - 1 \) is positive, \( y \) will be positive.
- If \( x - 1 \) is negative, \( y \) will be negative.

### Step 3: Check each option
1. **Option A: \( y = 0 \)**
- If \( y = 0 \), then \( \frac{1}{x - 1} = 0 \).
- This implies \( 1 = 0 \), which is not possible.
- Therefore, \( y \) cannot be \( 0 \).

2. **Option B: \( y = 1 \)**
- If \( y = 1 \), then \( \frac{1}{x - 1} = 1 \).
- This implies \( x - 1 = 1 \) or \( x = 2 \), which is valid.

3. **Option C: \( y = 2 \)**
- If \( y = 2 \), then \( \frac{1}{x - 1} = 2 \).
- This implies \( x - 1 = \frac{1}{2} \) or \( x = \frac{3}{2} \), which is valid.

4. **Option D: \( y = 3 \)**
- If \( y = 3 \), then \( \frac{1}{x - 1} = 3 \).
- This implies \( x - 1 = \frac{1}{3} \) or \( x = \frac{4}{3} \), which is valid.

### Conclusion
The only value that cannot be achieved by \( y \) is \( 0 \).

Thus, the answer is:
**A: 0**