Simplify each complex fraction. [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. Find a common denominator for the expressions in the numerator and denominator. Since both contain terms with \(x\) in the denominator, multiply numerator and denominator by \(x\) to eliminate the fractions within the fraction.
3. Multiply numerator and denominator by \(x\): \(\frac{x \left(1 + \frac{1}{x}\right)}{x \left(1 - \frac{1}{x}\right)}\).
4. Simplify inside the parentheses after multiplication: numerator becomes \(x \cdot 1 + x \cdot \frac{1}{x} = x + 1\), denominator becomes \(x \cdot 1 - x \cdot \frac{1}{x} = x - 1\).
5. Rewrite the complex fraction as a simple fraction: \(\frac{x + 1}{x - 1}\). This is the simplified form of the original complex fraction.

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.

Least Common Denominator (LCD)

The least common denominator is the smallest common multiple of the denominators in a set of fractions. Multiplying numerator and denominator by the LCD helps clear fractions within fractions, making the expression easier to simplify.

Algebraic Simplification

Algebraic simplification involves combining like terms, factoring, and reducing expressions to their simplest form. After clearing complex fractions, simplifying the resulting rational expression is essential to reach the final simplified form.