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Chapter 4: Problem 216

Express \(\frac{1}{x-3}+\frac{2}{x-2}\) as a single fraction.

Short Answer

Expert verified
\( \frac{1}{x-3} + \frac{2}{x-2} = \frac{3x-8}{(x-3)(x-2)}\)

Step by step solution

01

Identify the Least Common Denominator (LCD)

The denominators in the problem are \(x-3\) and \(x-2\). To combine the fractions, find the LCD, which is \((x-3)(x-2)\).
02

Rewrite Each Fraction with the LCD as the Denominator

Change each fraction so that they both have the common denominator \((x-3)(x-2)\). This involves multiplying each fraction by the appropriate term. The first fraction becomes: \(\frac{1}{x-3} \times \frac{x-2}{x-2} = \frac{x-2}{(x-3)(x-2)}\).The second fraction becomes: \(\frac{2}{x-2} \times \frac{x-3}{x-3} = \frac{2(x-3)}{(x-3)(x-2)}\).
03

Combine the Fractions

Now that both fractions have the same denominator, combine them: \( \frac{x-2}{(x-3)(x-2)} + \frac{2(x-3)}{(x-3)(x-2)} = \frac{x-2 + 2(x-3)}{(x-3)(x-2)}\).
04

Simplify the Numerator

Distribute and combine like terms in the numerator: \( x-2 + 2(x-3) = x-2 + 2x - 6 = 3x - 8\). Thus, the combined fraction becomes: \( \frac{3x-8}{(x-3)(x-2)}\).

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Key Concepts

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

least common denominator
Understanding the Least Common Denominator (LCD) is crucial when working with fractions, especially if their denominators are different.
The LCD is the smallest multiple that two or more denominators share. In our example, we have the fractions \( \frac{1}{x-3} \) and \( \frac{2}{x-2} \).
Since both factors are already in their simplest form, the LCD is simply the product of these denominators, which gives us \( (x-3)(x-2) \).
You can think of the LCD as creating a common 'space' for both fractions, enabling us to combine them effectively.
combining fractions
Once you've identified the LCD, the next step is to rewrite each fraction so they both share this common denominator. This process might sound tricky, but it simplifies significantly with practice.
For instance, converting \( \frac{1}{x-3} \) and \( \frac{2}{x-2} \) involves:
  • Multiplying \( \frac{1}{x-3} \) by \( \frac{x-2}{x-2} \) to get \( \frac{x-2}{(x-3)(x-2)} \).
  • Multiplying \( \frac{2}{x-2} \) by \( \frac{x-3}{x-3} \) to achieve \( \frac{2(x-3)}{(x-3)(x-2)} \).
By adjusting both fractions to share the LCD, combining them becomes straightforward: simply add their numerators together while maintaining the common denominator.
simplifying numerators
The next focus is on simplifying the numerators of the fractions we've combined. With both fractions adjusted for the LCD, our problem looks like: \( \frac{x-2}{(x-3)(x-2)} + \frac{2(x-3)}{(x-3)(x-2)} \).
Adding these fractions involves summing up their numerators: \( x-2 + 2(x-3) \).
Simplify by distributing and combining like terms:
  • Unwrap \( 2(x-3) \) to get \( 2x-6 \).
  • Combine: \( x-2 + 2x-6 = 3x-8 \).
This results in a single, simplified numerator for our combined fraction.
rational expressions
Finally, we are working with rational expressions here. A rational expression is a fraction where both the numerator and the denominator are polynomials.
In our exercise, we managed to simplify: \( \frac{1}{x-3} + \frac{2}{x-2} \) into a single rational expression: \( \frac{3x-8}{(x-3)(x-2)} \).
Understanding rational expressions is essential because they frequently appear in higher-level math.
They function similarly to regular fractions, but with polynomials in their place, adding layers of complexity but also enabling powerful algebraic manipulations.

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