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

Verified step by step guidance

1. Identify the complex fraction given: \(\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}\).
2. Rewrite the numerator and denominator to have a common denominator \(x\): numerator becomes \(\frac{x}{x} + \frac{1}{x} = \frac{x + 1}{x}\), denominator becomes \(\frac{x}{x} - \frac{1}{x} = \frac{x - 1}{x}\).
3. Express the complex fraction as a division of two fractions: \(\frac{\frac{x + 1}{x}}{\frac{x - 1}{x}}\).
4. Simplify by multiplying the numerator by the reciprocal of the denominator: \(\frac{x + 1}{x} \times \frac{x}{x - 1}\).
5. Cancel the common factor \(x\) in numerator and denominator to get the simplified expression: \(\frac{x + 1}{x - 1}\).

Key Concepts

Complex Fractions

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

Algebraic Manipulation

Algebraic manipulation includes operations like addition, subtraction, multiplication, and division of expressions involving variables. It is essential to carefully combine like terms and apply arithmetic rules to simplify expressions, especially when variables appear in denominators.

Reciprocal and Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. Understanding this allows one to simplify complex fractions by converting division into multiplication, making the expression easier to handle and reduce.