To simplify the expression \( \frac{x}{-\frac{1}{x} - 1} \), let's proceed step by step.

### Step 1: Simplify the denominator
The denominator is \( -\frac{1}{x} - 1 \). To combine these terms, find a common denominator:

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

So the expression becomes:

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

### Step 2: Simplify the expression
Dividing by a fraction is the same as multiplying by its reciprocal:

\[
\frac{x}{-\frac{1 + x}{x}} = x \times \left(-\frac{x}{1 + x}\right)
\]

### Step 3: Final simplification
Now, multiply:

\[
-x \times \frac{x}{1 + x} = -\frac{x^2}{1 + x}
\]

Thus, the simplified form of the expression is:

\[
-\frac{x^2}{1 + x}
\]

Would you like further details or have any questions?

Here are five related questions to explore:
1. How do you simplify expressions involving complex fractions?
2. What are common strategies for finding a common denominator?
3. Can you explain how to handle negative signs in complex algebraic expressions?
4. How does multiplying by the reciprocal simplify division of fractions?
5. What happens when \( x = -1 \) in the simplified expression?

**Tip:** Always check for any values that might make the denominator zero, as they are excluded from the domain of the expression.

Learn how to simplify the fraction x divided by negative one over x minus one step-by-step. This problem involves algebraic fraction simplification and is suitable for students in Grades 9-12.

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