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Chapter 6: Problem 25

Simplify each complex fraction. $$\frac{1-\frac{1}{y-1}}{3+\frac{1}{y+1}}$$

Short Answer

Expert verified
The simplified form is: \(\frac{(y-2)(y+1)}{(y-1)(3y+4)}\)

Step by step solution

01

Rewrite the Complex Fraction

Identify and rewrite the numerator and the denominator separately. We have \(1 - \frac{1}{y-1}\) in the numerator and \(3 + \frac{1}{y+1}\) in the denominator.
02

Simplify the Numerator

Find a common denominator for \(1 - \frac{1}{y-1}\). The common denominator for \(1\) and \(\frac{1}{y-1}\) is \(y-1\). Rewrite the expression as: \[ 1 - \frac{1}{y-1} = \frac{y-1}{y-1} - \frac{1}{y-1} = \frac{y-2}{y-1} \]
03

Simplify the Denominator

Similarly, find a common denominator for \(3 + \frac{1}{y+1}\). The common denominator for \(3\) and \(\frac{1}{y+1}\) is \(y+1\). Rewrite the expression as: \[ 3 + \frac{1}{y+1} = \frac{3(y+1)}{y+1} + \frac{1}{y+1} = \frac{3y + 3 + 1}{y+1} = \frac{3y + 4}{y+1} \]
04

Form the New Fraction and Simplify

Now, we have the fraction: \[ \frac{\frac{y-2}{y-1}}{\frac{3y+4}{y+1}} \] To simplify, multiply by the reciprocal of the denominator: \[ \frac{y-2}{y-1} \times \frac{y+1}{3y+4} = \frac{(y-2)(y+1)}{(y-1)(3y+4)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
Understanding common denominators is crucial when working with fractions, especially complex ones. A common denominator is a shared multiple of the denominators of two or more fractions. To simplify fractions like in the given exercise, you need to find a common denominator so you can combine them easily. For instance, when combining 1 and \(\frac{1}{y-1}\), the common denominator is \(y-1\). This transforms the expression into:

\[\frac{y-1}{y-1} - \frac{1}{y-1}\]

This step is necessary because it allows you to combine the terms into a single fraction. Once you have a common denominator, you can add or subtract the fractions by working with their numerators.
Reciprocal
The concept of a reciprocal is vital for simplifying complex fractions. The reciprocal of a fraction simply means flipping the numerator and the denominator. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). In the exercise, after forming the new fraction:

\[\frac{\frac{y-2}{y-1}}{\frac{3y+4}{y+1}}\]

you take the reciprocal of the denominator to simplify:

\[\frac{y-2}{y-1} \times \frac{y+1}{3y+4}\]

This step transforms division into multiplication, making it easier to simplify the resulting fraction.
Fraction Simplification
Simplifying fractions helps you to work with simpler expressions. The goal is to reduce the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. In the final step of the provided exercise:

\[\frac{y-2}{y-1} \times \frac{y+1}{3y+4} = \frac{(y-2)(y+1)}{(y-1)(3y+4)}\]

Check if any factors in the numerator and the denominator cancel each other out. If they do, remove these common factors to simplify the fraction further. Fraction simplification often involves factorizing and canceling common terms to reach the simplest form possible.

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Most popular questions from this chapter

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{3 x}{5}}{y}$$

Find the solution set to each equation. $$\frac{x+5}{2}=\frac{3}{x}$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{5}{h}=\frac{h}{5}$$

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{2 x}=\frac{3}{2 x}$$

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{2 a}+x^{a}-6}{x^{2 a}+6 x^{a}+9} \div \frac{x^{2 a}-4}{x^{2 a}+2 x^{a}-3}$$
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