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Chapter 6: Problem 72

Add or subtract as indicated. $$\frac{-2}{x-1}+\frac{2}{x+1}$$

Short Answer

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

Step by step solution

01

Find the Common Denominator

To add or subtract fractions, the denominators must be the same. The first fraction has the denominator \(x-1\) and the second fraction has the denominator \(x+1\). The common denominator will be the product of these two, so the common denominator is \((x-1)(x+1)\).
02

Rewrite Each Fraction with the Common Denominator

Rewrite each fraction so they both have the common denominator. For the first fraction: \[\frac{-2}{x-1} = \frac{-2(x+1)}{(x-1)(x+1)}\] For the second fraction: \[\frac{2}{x+1} = \frac{2(x-1)}{(x-1)(x+1)}\]
03

Combine the Numerators

Now that both fractions have the same denominator, combine the numerators: \[\frac{-2(x+1) + 2(x-1)}{(x-1)(x+1)}\]
04

Simplify the Numerator

Expand and simplify the numerator: \[\frac{-2x - 2 + 2x - 2}{(x-1)(x+1)} = \frac{-4}{(x-1)(x+1)}\]

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Key Concepts

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

common denominators
When working with fractions, adding and subtracting them requires a common denominator. The common denominator is a shared multiple of each fraction's denominator.
In our problem, the denominators are \(x-1\) and \(x+1\). To find a common denominator, we multiply these denominators together.
This method ensures both fractions will share the same base, making it possible to combine them.
For example, the common denominator of \[\frac{-2}{x-1} + \frac{2}{x+1}\] is \((x-1)(x+1)\). By rewriting the fractions with this common denominator, we set them up for addition or subtraction.
addition and subtraction of fractions
With a common denominator in place, the next step is adding or subtracting the numerators.
Here's how it works in our exercise:
  • We rewrite \[\frac{-2}{x-1}\] and \[\frac{2}{x+1}\] using the common denominator \((x-1)(x+1)\).
  • Thus, \[\frac{-2}{x-1} \ \text{becomes} \ \frac{-2(x+1)}{(x-1)(x+1)}\] and \[\frac{2}{x+1} \ \text{becomes} \ \frac{2(x-1)}{(x-1)(x+1)}\].
  • Now, we can combine the numerators: \[-2(x+1) + 2(x-1)\] over the common denominator \((x-1)(x+1)\).
Once the fractions have the same denominator, the numerators are added or subtracted as usual.
simplifying algebraic expressions
After combining numerators, the next critical step is simplifying the algebraic expression.
We merged our fractions into \[-2(x+1) + 2(x-1)\] over \((x-1)(x+1)\).
It's time to simplify the numerator:
  • Distribute the coefficients: \[-2x - 2 + 2x - 2\].
  • Combine like terms: Here, including \(-2x\) and \(+2x\) cancels each other out, leaving us with \-2 - 2\ or \-4\.
Thus, the simplified expression is \[\frac{-4}{(x-1)(x+1)}\].
Simplifying algebraic expressions helps in solving complex problems and makes the solutions more manageable.

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Most popular questions from this chapter

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{5}{9} \cdot \frac{12}{25} $$

Add or subtract as indicated. $$\frac{8}{(3 r-1)^{2}}+\frac{2}{3 r-1}-6$$

Multiply or divide as indicated. $$ \frac{9 s-12 t}{2 s+2 t} \cdot \frac{3 s+3 t}{4 t-3 s} $$

Write each rational expression in lowest terms. $$ \frac{5 k-10}{20-10 k} $$

Add or subtract as indicated. $$\frac{6 x+5 y}{6 x^{2}+5 x y-4 y^{2}}-\frac{x+2 y}{9 x^{2}-16 y^{2}}$$
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