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Chapter 5: Problem 13

Find all complex solutions for each equation by hand. $$\frac{2 x}{x^{2}-1}=\frac{2}{x+1}-\frac{1}{x-1}$$

Short Answer

Expert verified
The solution is \(x = -3\).

Step by step solution

01

Find a common denominator

To solve the equation \(\frac{2 x}{x^{2}-1}=\frac{2}{x+1}-\frac{1}{x-1}\), we first recognize that \(x^2 - 1\) can be factored as \((x-1)(x+1)\). Therefore, the common denominator for the entire equation is \((x-1)(x+1)\). We will use this to rewrite each term under a common denominator.
02

Rewrite the equation using the common denominator

Rewrite each fraction using the common denominator:- The left side becomes \(\frac{2x}{(x-1)(x+1)}\).- The right side's first fraction is \(\frac{2(x-1)}{(x-1)(x+1)}\).- The right side's second fraction is \(\frac{1(x+1)}{(x-1)(x+1)}\).Thus, the equation becomes: \[ \frac{2x}{(x-1)(x+1)} = \frac{2(x-1)}{(x-1)(x+1)} - \frac{1(x+1)}{(x-1)(x+1)} \].
03

Combine the right side

Combine the fractions on the right side over the common denominator:\[ \frac{2(x-1) - (x+1)}{(x-1)(x+1)} = \frac{2x - 2 - x - 1}{(x-1)(x+1)} = \frac{x - 3}{(x-1)(x+1)} \].Now the equation is:\[ \frac{2x}{(x-1)(x+1)} = \frac{x - 3}{(x-1)(x+1)} \].
04

Clear the denominators

Since both sides of the equation now have the same denominator, we can set the numerators equal to each other:\(2x = x - 3\).
05

Solve for x

Isolate \(x\) by subtracting \(x\) from both sides:\(2x - x = -3\)This simplifies to:\(x = -3\).
06

Verify the solution

Substitute \(x = -3\) back into the original equation to check for any extraneous solutions:- Left side: \(\frac{2(-3)}{(-3)^2-1} = \frac{-6}{8}\).- Right side: \(\frac{2}{-2} - \frac{1}{-4} = -1 + \frac{1}{4} = -\frac{3}{4}\).Both sides equal \(-\frac{3}{4}\), confirming \(x = -3\) is a valid solution.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a crucial step in simplifying complex equations. When you encounter a polynomial such as \(x^2 - 1\), it can often be rewritten to make solving the equation easier.
In this case, \(x^2 - 1\) can be factored into \((x-1)(x+1)\).
  • Factoring helps in identifying common roots and expressing the polynomial in a more manageable form.
  • This technique simplifies operations like addition, subtraction, or finding common denominators of algebraic fractions.
Grasping this concept is essential since it often forms the backbone of many algebraic solutions.
Common Denominator
Finding a common denominator is a pivotal step in solving equations that involve fractions. In the given equation, \(\frac{2 x}{x^{2}-1} = \frac{2}{x+1} - \frac{1}{x-1}\), each side has different denominators initially.
To solve, you need to convert these into a common denominator to manage the fractions effectively.
  • The common denominator here is \((x-1)(x+1)\), which simplifies the equation handling.
  • Doing so ensures that each fraction can be combined or compared easily.
This simplification is fundamental because matched denominators let us equate or combine fractions in a unified form.
Cross-Multiplication
Once the fractions have a common denominator on both sides of an equation, you can effectively clear these denominators using cross-multiplication. Here, it involves setting the numerators equal since the denominators are already the same.
This method transforms \(\frac{2x}{(x-1)(x+1)} = \frac{x - 3}{(x-1)(x+1)}\) into the simpler form of equation, \(2x = x - 3\).
  • This step is pivotal because it converts a complex fraction into a straightforward linear equation.
  • With the numerators isolated, solving becomes a matter of basic algebra, enhancing clarity and focus on solving for the variable.
Understanding this approach helps in tackling a broad array of algebraic problems more efficiently.
Verification of Solutions
After calculating potential solutions, verifying them is key to ensuring accuracy. When \(x = -3\) was found as a solution, it was necessary to substitute it back into the original equation to confirm its validity.
Here's how:
  • Substituting \(x = -3\) into both sides yielded \(-\frac{3}{4}\) as the result, confirming the solution.
  • This verification step checks for any errors in computation or extraneous solutions that may not satisfy the original equation.
Such verification is a vital part of problem-solving, providing a reliable conclusion to your algebraic journey.

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Most popular questions from this chapter

Use long division of polynomials to show that for $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{2}+x-12}$$ if we divide the numerator by the denominator, then the quotient polynomial is \(x^{2}-x+8,\) and the remainder is \(-20 x+100 .\) Graph both \(f(x)\) and \(g(x)=x^{2}-x+8\) in the window \([-50,50]\) by \([0,1000] .\) Comment on the appearance of the two graphs. Explain how the graph of \(f\) approaches that of \(g\) as \(|x| \rightarrow \infty\).

Insect Population Suppose that an insect population in millions is modeled by $$ f(x)=\frac{10 x+1}{x+1} $$ where \(x \geq 0\) is in months. (a) Graph \(f\) in the window \([0,14]\) by \([0,14] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population after several months? (d) Interpret the horizontal asymptote.

Construction Find possible dimensions for a closed box with volume 196 cubic inches, surface area 280 square inches, and length that is twice the width.

Solve each problem involving rate of work. A couple is laying a tile floor. Working alone, one can do the job in 20 hours. If the two of them work together, they can complete the job in 12 hours. How long would it take the other one to lay the floor working alone?

Solve each problem involving rate of work. Two grandparents want to pick up the mess that their granddaughter has made in her playroom. One can do it in 15 minutes working alone. The other, working alone, can clean it in 12 minutes. How long will it take them if they work together?
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