To simplify the expression 

\[
\frac{\frac{1}{x} - 1}{x - 1},
\]

follow these steps:

1. **Simplify the numerator**:

   The numerator \(\frac{1}{x} - 1\) can be rewritten with a common denominator:

   \[
   \frac{1}{x} - 1 = \frac{1 - x}{x}.
   \]

   So the expression becomes:

   \[
   \frac{\frac{1 - x}{x}}{x - 1}.
   \]

2. **Simplify the complex fraction**:

   The expression \(\frac{\frac{1 - x}{x}}{x - 1}\) can be simplified by multiplying the numerator by the reciprocal of the denominator:

   \[
   \frac{1 - x}{x} \times \frac{1}{x - 1} = \frac{1 - x}{x(x - 1)}.
   \]

3. **Simplify the expression further**:

   Notice that \(1 - x\) is equivalent to \(-(x - 1)\), so:

   \[
   \frac{-(x - 1)}{x(x - 1)}.
   \]

   The \((x - 1)\) terms cancel out, leaving:

   \[
   \frac{-1}{x}.
   \]

Thus, the simplified form of the expression is:

\[
-\frac{1}{x}.
\]

Would you like to explore any specific aspect of this simplification further?

Here are some related questions you might consider:
1. What is the limit of this expression as \(x\) approaches 0?
2. How does this expression compare to the original form in terms of graphing?
3. What if the original numerator was \(\frac{1}{x} + 1\)?
4. How would you integrate this expression with respect to \(x\)?
5. What are the implications of this expression in a real-world application?

**Tip:** When simplifying expressions, always look for opportunities to factor and cancel out common terms to reduce the complexity of the expression.

Learn how to simplify the expression (1/x - 1)/(x - 1) step-by-step with detailed explanations and examples. This problem involves algebraic fraction simplification and explores the concept of limits as x approaches 0. Suitable for advanced high school students and anyone interested in algebraic simplification.

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