Simplify each complex fraction. See Examples 5 and 6. (1/(x + 1) − 1/x) / (1/x)

Verified step by step guidance

1. Rewrite the complex fraction clearly to understand its structure. The expression is: \(\frac{\frac{1}{x+1} - \frac{1}{x}}{\frac{1}{x}}\).
2. Focus on simplifying the numerator first: \(\frac{1}{x+1} - \frac{1}{x}\). Find a common denominator for these two fractions, which is \(x(x+1)\).
3. Rewrite each fraction in the numerator with the common denominator: \(\frac{x}{x(x+1)} - \frac{x+1}{x(x+1)}\).
4. Combine the fractions in the numerator by subtracting the numerators over the common denominator: \(\frac{x - (x+1)}{x(x+1)}\).
5. Simplify the numerator of the combined fraction and then divide this result by the denominator \(\frac{1}{x}\) by multiplying by its reciprocal.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying numerator and denominator by the least common denominator to eliminate smaller fractions.

Algebraic Manipulation

Algebraic manipulation includes operations like factoring, expanding, and simplifying expressions. In this context, it is essential to carefully handle variables and expressions in both numerator and denominator to combine terms correctly and reduce the fraction to its simplest form.

Fraction Division and Multiplication

Dividing fractions involves multiplying by the reciprocal of the divisor. When simplifying complex fractions, converting division into multiplication by the reciprocal helps to combine and simplify the expression efficiently.