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Chapter 7: Problem 67

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{4 x}{x-1}-\frac{2}{x+1}-\frac{4}{x^{2}-1} $$

Short Answer

Expert verified
\( \frac{4x^2 + 2x - 2}{(x-1)(x+1)} \)

Step by step solution

01

- Recognize the common denominator

Identify the common denominator for the expression: \( (x-1), (x+1), (x^2-1) \). Notice that \( x^2-1 = (x-1)(x+1) \). So, the common denominator is \( (x-1)(x+1) \).
02

- Rewrite each fraction

Rewrite each fraction with the common denominator \( (x-1)(x+1) \). \[ \frac{4x}{x-1} = \frac{4x(x+1)}{(x-1)(x+1)} \] \[ \frac{2}{x+1} = \frac{2(x-1)}{(x+1)(x-1)} \] \[ - \frac{4}{x^2-1} = - \frac{4}{(x-1)(x+1)} \]
03

- Combine all fractions

Combine all fractions over the common denominator \( (x-1)(x+1) \): \[ \frac{4x(x+1) - 2(x-1) - 4}{(x-1)(x+1)} \]
04

- Simplify the numerator

Expand and simplify the numerator: \[ 4x(x+1) = 4x^2 + 4x \] \[ -2(x-1) = -2x + 2 \] \[ 4x^2 + 4x - 2x + 2 - 4 = 4x^2 + 2x - 2 \]
05

- Write the final expression

Combine the simplified numerator and denominator: \[ \frac{4x^2 + 2x - 2}{(x-1)(x+1)} \]. This is already in its lowest terms.

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Key Concepts

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

Common Denominator
When working with algebraic fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of all fractions involved. This allows us to easily add or subtract the fractions. In this exercise, the denominators are \(x-1\), \(x+1\), and \(x^2-1\). Notice that \(x^2-1\) can be factored into \((x-1)(x+1)\). Thus, the least common denominator (LCD) is \((x-1)(x+1)\). This step sets the stage for rewriting each fraction with the same denominator, making it possible to combine them.
Fraction Addition and Subtraction
Once we have a common denominator, we can rewrite and combine the fractions. Let's break it down using our example:
1. Rewrite each fraction with the denominator \((x-1)(x+1)\): \( \frac{4x}{x-1} = \frac{4x(x+1)}{(x-1)(x+1)} \), \( \frac{2}{x+1} = \frac{2(x-1)}{(x+1)(x-1)} \), and the third term is already \( \frac{4}{(x-1)(x+1)} \).
2. Combine these into one fraction: \( \frac{4x(x+1) - 2(x-1) - 4}{(x-1)(x+1)} \).
This step-by-step rewriting and combining of fractions allows us to handle multiple fractions as a single entity for further simplification.
Simplifying Rational Expressions
To simplify a rational expression, we focus on simplifying the numerator and then ensuring the result is in its lowest terms.
First, expand the numerator:
  • \( 4x(x+1) = 4x^2 + 4x \)
  • \( -2(x-1) = -2x + 2 \)
Combine these: \(4x^2 + 4x - 2x + 2 - 4 = 4x^2 + 2x - 2 \).
By doing this, we achieve the simplified expression: \( \frac{4x^2 + 2x - 2}{(x-1)(x+1)} \). No further factorization or reduction is possible here, so our expression remains in its lowest terms. Simplifying rational expressions involves careful expansion and combination to reach a more straightforward form.

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Most popular questions from this chapter

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5 x-y}{x^{2}+x y-2 y^{2}}-\frac{3 x+2 y}{x^{2}+5 x y-6 y^{2}} $$

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