Simplify each complex fraction. See Examples 5 and 6. 1 1 ——— - —— x + 1 x———————— 1 —— x

Verified step by step guidance

1. Step 1: Identify the complex fraction: \( \frac{\frac{1}{x+1} - \frac{1}{x}}{\frac{1}{x}} \).
2. Step 2: Simplify the numerator: \( \frac{1}{x+1} - \frac{1}{x} \). Find a common denominator for the fractions in the numerator, which is \( x(x+1) \).
3. Step 3: Rewrite each fraction in the numerator with the common denominator: \( \frac{x}{x(x+1)} - \frac{x+1}{x(x+1)} \).
4. Step 4: Combine the fractions in the numerator: \( \frac{x - (x+1)}{x(x+1)} = \frac{x - x - 1}{x(x+1)} = \frac{-1}{x(x+1)} \).
5. Step 5: Simplify the entire complex fraction: \( \frac{\frac{-1}{x(x+1)}}{\frac{1}{x}} \). Multiply by the reciprocal of the denominator: \( \frac{-1}{x(x+1)} \times x = \frac{-x}{x(x+1)} \). Simplify to get \( \frac{-1}{x+1} \).

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 rewrites the complex fraction as a single fraction. This process often involves algebraic manipulation to eliminate the nested fractions.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When simplifying complex fractions, identifying the least common denominator (LCD) allows for the combination of fractions into a single fraction. This step is crucial for simplifying the overall expression and making calculations more manageable.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes operations such as factoring, distributing, and combining like terms. In the context of complex fractions, effective algebraic manipulation is essential for simplifying the expression and achieving a clearer, more concise form.