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Chapter 0: Problem 52

Find each sum or difference. $$\frac{2}{x-1}+\frac{1}{1-x}$$

Short Answer

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

Step by step solution

01

Simplify the fractions

Notice that the second fraction, \( \frac{1}{1-x} \), can be rewritten as \( \frac{-1}{x-1} \) since \( 1-x \) is equal to \( -(x-1) \). This allows us to rewrite the expression as \(\frac{2}{x-1} - \frac{1}{x-1}\).
02

Combine the fractions

Now that both fractions have the same denominator, \( x-1 \), we can combine them into a single fraction: \(\frac{2 - 1}{x-1} = \frac{1}{x-1}\).
03

Simplify the result

The expression \( \frac{1}{x-1} \) is already in its simplest form, so no further simplification is needed.

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Key Concepts

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

Understanding Fractions
Fractions and rational expressions are fundamental concepts in algebra. A fraction consists of a numerator and a denominator, much like a division problem. In a rational expression, rather than just numbers, we are dealing with polynomials in the numerator and/or the denominator.

In the problem provided, we had two fractions: \( \frac{2}{x-1} \) and \( \frac{1}{1-x} \). Although these may at first appear different due to their denominators, understanding fractions helps us see through this. These denominators are negatives of each other, as \( 1-x = -(x-1) \).

  • The numerator is the top part of the fraction, which signifies how many parts we have.
  • The denominator is the bottom part, telling us into how many parts the whole is divided.
  • Fractions represent division: the numerator divided by the denominator.
Finding a Common Denominator
Finding a common denominator is crucial when adding or subtracting fractions. It's the shared base that allows us to combine fractions into one. Without a common denominator, the fractions are incompatible for direct addition or subtraction.

In our example, once we address that \( 1-x \) can be rewritten as \(-(x-1)\), the fractions \( \frac{2}{x-1} \) and \( \frac{-1}{x-1} \) emerge with a common denominator of \( x-1 \). This transformation simplifies our task significantly.

  • Identify potential common factors in the denominators.
  • Rewrite fractions to reflect their equivalent forms.
  • Ensure previously negative denominators are handled correctly to avoid errors.
Matching the denominators allows us smooth integration, turning the problem into a simple subtraction.
Simplification Steps
Simplifying fractions or rational expressions is the process of making an expression as concise as possible. In our problem, after rewriting the fractions to have a common denominator, simplification came naturally.

Having the fraction \( \frac{2}{x-1} - \frac{1}{x-1} \), combine by subtracting the numerators to get \( \frac{2-1}{x-1} = \frac{1}{x-1} \).

  • Subtract the numerators directly since denominators are the same.
  • Ensure the result is expressed in its simplest form.
  • Simplification is complete when no further reduction is possible.
The result \( \frac{1}{x-1} \) is already simplified, illustrating the elegance of the simplification steps which condense complexity into clarity.

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Most popular questions from this chapter

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{45}$$

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