Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Verified step by step guidance

1. Identify the complex fraction: the numerator is \(\frac{y+3}{y} - \frac{4}{y-1}\) and the denominator is \(\frac{y}{y-1} + \frac{1}{y}\).
2. Find a common denominator for the numerator terms, which are \(\frac{y+3}{y}\) and \(\frac{4}{y-1}\). The common denominator is \(y(y-1)\). Rewrite each fraction with this common denominator:
3. \[\frac{(y+3)(y-1)}{y(y-1)} - \frac{4y}{y(y-1)}.\]
4. Combine the numerator fractions into a single fraction by subtracting the numerators over the common denominator:
5. \[\frac{(y+3)(y-1) - 4y}{y(y-1)}.\]
6. Similarly, find a common denominator for the denominator terms \(\frac{y}{y-1}\) and \(\frac{1}{y}\), which is also \(y(y-1)\). Rewrite each fraction:
7. \[\frac{y^2}{y(y-1)} + \frac{y-1}{y(y-1)}.\]
8. Combine the denominator fractions into a single fraction by adding the numerators over the common denominator:
9. \[\frac{y^2 + (y-1)}{y(y-1)}.\]
10. Now, rewrite the original complex fraction as a division of two single fractions:
11. \[\frac{\frac{(y+3)(y-1) - 4y}{y(y-1)}}{\frac{y^2 + (y-1)}{y(y-1)}} = \frac{(y+3)(y-1) - 4y}{y(y-1)} \div \frac{y^2 + (y-1)}{y(y-1)}.\]
12. Recall that dividing by a fraction is the same as multiplying by its reciprocal. So rewrite the expression as:
13. \[\frac{(y+3)(y-1) - 4y}{y(y-1)} \times \frac{y(y-1)}{y^2 + (y-1)}.\]
14. Cancel the common factors \(y(y-1)\) in numerator and denominator, then simplify the remaining expressions by expanding and combining like terms.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by combining the smaller fractions in the numerator and denominator before dividing.

Finding a Common Denominator

To add or subtract fractions, you must find a common denominator, which is a shared multiple of the denominators involved. This allows you to rewrite each fraction with the same denominator, making it possible to combine them into a single fraction.

Dividing Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. After simplifying the numerator and denominator separately, you divide the two fractions by multiplying the numerator by the reciprocal of the denominator.