https://file.aiquickdraw.com/mathbot/file/1725737677141pqpr2sxl.jpg

The problem presented involves the equation:

\[
|x - 1| = 0, \quad \forall x \in (0, 1)
\]

### Let's break this down:

1. **Absolute Value Property**: The absolute value equation \( |x - 1| = 0 \) implies that the expression inside the absolute value must be zero. That is:
   \[
   x - 1 = 0 \implies x = 1
   \]
2. **Domain Restriction**: The condition \( \forall x \in (0, 1) \) means that \( x \) is restricted to values between 0 and 1, **excluding** 1.

### Analysis:
- The equation \( |x - 1| = 0 \) is **only true when \( x = 1 \)**.
- However, \( x = 1 \) is **not** in the interval \( (0, 1) \), as the interval is open and excludes the endpoint 1.

Therefore, the equation \( |x - 1| = 0 \) cannot be true for any \( x \) in \( (0, 1) \), making the statement **false**.

### Conclusion:
The correct choice is "False."

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Would you like any further explanation or details? Here are some related questions to deepen understanding:

1. What is the definition of an open interval in set theory?
2. How would the answer change if the interval were closed, \( [0, 1] \)?
3. What are the general properties of absolute value equations?
4. How does the equation \( |x - 1| = 0 \) behave over different intervals?
5. Can we find any other intervals where \( |x - 1| = 0 \) holds?

**Tip**: Remember, absolute value equations are useful for finding distances from a reference point!

Understand the solution to the absolute value equation |x - 1| = 0 within the open interval (0,1). This problem discusses key concepts such as absolute value properties and interval restrictions, making it suitable for high school students (Grades 9-12).

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