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The given problem asks to find the vertical asymptotes of the function:

\[
y = (x + 1) \cdot \ln\left(2 - \frac{1}{x}\right)
\]

### To solve for the vertical asymptotes:

1. **Look at the logarithmic function**: For the natural logarithm function \(\ln\), the argument must be strictly positive. So, we need to solve:

\[
2 - \frac{1}{x} > 0
\]

\[
2 > \frac{1}{x} \quad \Rightarrow \quad x > \frac{1}{2}
\]

Thus, \(x = \frac{1}{2}\) is a critical point.

2. **Check for undefined points**: The logarithmic function becomes undefined if \(x = 0\), because \(\frac{1}{x}\) would become infinite.

Thus, the vertical asymptotes occur at:

- \(x = 0\) (because the logarithmic term is undefined at this point).
- \(x = \frac{1}{2}\) (due to the boundary condition).

### Conclusion:
The vertical asymptotes are at \(x = 0\) and \(x = \frac{1}{2}\). Therefore, the correct answer is **d. \(x = 0\) and \(x = \frac{1}{2}\)**.

Would you like further explanation or details?

Here are 5 related questions to explore:
1. How do you determine horizontal asymptotes for a given function?
2. How does the natural logarithm function behave near its vertical asymptotes?
3. What happens to the function \(y = \ln(x)\) as \(x\) approaches 0 from the right?
4. Can a function have multiple vertical asymptotes?
5. What role do asymptotes play in curve sketching?

**Tip**: When working with logarithmic functions, always check the domain carefully to identify where the argument becomes non-positive, as that will give clues to asymptotes or undefined points.

Find the vertical asymptotes of the following function: y = (x + 1) ln(2 - 1/x).

This problem demonstrates how to find vertical asymptotes for the function y = (x + 1) ln(2 - 1/x). The key concepts involve analyzing logarithmic functions and identifying points where the function becomes undefined. Suitable for students in Grades 11-12.

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