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: \( \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} \).
2. To simplify, find a common denominator for the terms in the numerator and the denominator. Here, the common denominator is \( x \).
3. Rewrite the numerator: \( 1 + \frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x+1}{x} \).
4. Rewrite the denominator: \( 1 - \frac{1}{x} = \frac{x}{x} - \frac{1}{x} = \frac{x-1}{x} \).
5. Now, divide the rewritten numerator by the rewritten denominator: \( \frac{\frac{x+1}{x}}{\frac{x-1}{x}} = \frac{x+1}{x-1} \).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the inner fractions and then simplifies the overall expression. Understanding how to manipulate fractions is crucial for effectively simplifying complex fractions.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When simplifying complex fractions, finding a common denominator allows you to combine or compare the fractions more easily. This step is essential for reducing the complexity of the expression and achieving a simpler form.

Algebraic Manipulation

Algebraic manipulation involves applying algebraic rules to rearrange and simplify expressions. This includes operations such as factoring, distributing, and combining like terms. Mastery of these techniques is necessary for effectively simplifying complex fractions and solving algebraic equations.