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.