Simplify each complex fraction. See Examples 5 and 6. y + 3 4 ———— - ———— y y - 1—————————— y 1 ——— + —— y - 1 y

Verified step by step guidance

1. Identify the complex fraction structure: \( \frac{\frac{y + 3}{y} - \frac{4}{y - 1}}{\frac{y}{y - 1} + \frac{1}{y}} \).
2. Find a common denominator for the fractions in the numerator: The common denominator for \( \frac{y + 3}{y} \) and \( \frac{4}{y - 1} \) is \( y(y - 1) \).
3. Rewrite the fractions in the numerator with the common denominator: \( \frac{(y + 3)(y - 1) - 4y}{y(y - 1)} \).
4. Find a common denominator for the fractions in the denominator: The common denominator for \( \frac{y}{y - 1} \) and \( \frac{1}{y} \) is \( y(y - 1) \).
5. Rewrite the fractions in the denominator with the common denominator: \( \frac{y^2 + (y - 1)}{y(y - 1)} \).

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 and can help clarify the relationships between the variables involved.

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 for the combination of fractions into a single fraction, making calculations easier. This is particularly important in algebraic expressions where the denominators may be polynomials or other expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using various operations such as addition, subtraction, multiplication, and division. In the context of complex fractions, this may include factoring, distributing, and combining like terms. Mastery of these techniques is essential for effectively simplifying complex expressions and solving equations.