Simplify each complex fraction. [ 1/(x+1) - 1/x ] / (1/x)

Verified step by step guidance

1. Identify the complex fraction: \(\frac{\frac{1}{x+1} - \frac{1}{x}}{\frac{1}{x}}\).
2. Simplify the numerator by finding a common denominator for the two fractions: the common denominator is \(x(x+1)\), so rewrite each fraction as \(\frac{x}{x(x+1)} - \frac{x+1}{x(x+1)}\).
3. Combine the fractions in the numerator: \(\frac{x - (x+1)}{x(x+1)}\).
4. Simplify the numerator expression inside the fraction: \(x - (x+1) = x - x - 1 = -1\), so the numerator becomes \(\frac{-1}{x(x+1)}\).
5. Divide the simplified numerator by the denominator \(\frac{1}{x}\) by multiplying the numerator by the reciprocal of the denominator: \(\frac{-1}{x(x+1)} \times x\).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying complex fractions involves rewriting them as a single simple fraction by combining or dividing the inner fractions.

Common Denominator

To subtract or add fractions, you must find a common denominator. This allows you to combine the fractions into a single fraction by expressing each fraction with the same denominator.

Dividing Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. When simplifying complex fractions, after combining the numerator and denominator, you multiply the numerator by the reciprocal of the denominator to simplify the expression.