Problem 52 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{2}{x-1} - \frac{1}{x-1}\).

Step 2: Combine the fractions

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

Step 3: Simplify the result

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

Key Concepts

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.