To simplify the expression \(\frac{\left(\frac{1}{x} - 1\right)}{x - 1}\), let's break it down step by step:

1. **Combine the terms in the numerator**:
   \[
   \frac{1}{x} - 1 = \frac{1 - x}{x}
   \]
   So, the expression becomes:
   \[
   \frac{\frac{1 - x}{x}}{x - 1}
   \]

2. **Simplify the complex fraction**:
   \[
   \frac{1 - x}{x} \div (x - 1) = \frac{1 - x}{x(x - 1)}
   \]

3. **Simplify the expression further**:
   Notice that \(1 - x\) is equivalent to \(-(x - 1)\). So, we can rewrite the numerator:
   \[
   \frac{-(x - 1)}{x(x - 1)}
   \]
   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 delve deeper into this explanation or ask about another related topic?

Here are some questions you might explore next:
1. How does this expression behave as \(x\) approaches zero?
2. What happens to the expression as \(x\) approaches infinity?
3. How would you differentiate this expression with respect to \(x\)?
4. What if the denominator was \(x + 1\) instead of \(x - 1\)?
5. How could this expression be applied in a real-world context?

**Tip:** When simplifying complex fractions, always look for common factors that can be canceled out to simplify the expression further.

Learn how to simplify the expression (1/x - 1)/(x - 1) with a detailed step-by-step solution. This problem covers algebraic concepts related to fraction simplification. Suitable for students in Grades 9-12.

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