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Chapter 1: Problem 54

Perform the indicated operation and simplify. $$\frac{3}{x-2}+\frac{5}{2-x}$$

Short Answer

Expert verified
\(\frac{-2}{x-2}\)

Step by step solution

01

Recognize the Common Denominator

First, notice that the denominators \( x-2 \) and \( 2-x \) are closely related. Specifically, \( 2-x \) can be rewritten as \((-(x-2))\). Thus, the common denominator is \(-(x-2)\).
02

Adjust Fractions to Have Common Denominator

Rewrite the second fraction so both fractions have the same denominator of \(-(x-2)\). This gives: \(\frac{3}{x-2} + \frac{-5}{x-2}\).
03

Combine the Fractions

Now that both fractions have a common denominator, combine them:\[\frac{3}{x-2} + \frac{-5}{x-2} = \frac{3 + (-5)}{x-2}\] Simplifying the numerator gives \(-2\).
04

Simplify the Expression

Thus, the expression simplifies to: \(\frac{-2}{x-2}\).

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Key Concepts

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

Common Denominators
When dealing with rational expressions, finding a common denominator is crucial for adding or subtracting fractions. A common denominator is the shared bottom part of two or more fractions that allows for them to be combined. Here are some key points to remember about common denominators:
  • Identify the denominators of each fraction.
  • Rewrite expressions to achieve a common denominator if they are different.
  • Having a common denominator allows you to add or subtract the numerators directly.
In our exercise, the denominators were initially \(x-2\) and \(2-x\). They look different but can be seen as equivalent because \(2-x = -(x-2)\). Finding this equivalence allows you to recombine the fractions into a single expression.
Fraction Simplification
Once fractions have a common denominator, they can be combined, and further simplification may follow. This involves reducing the numerator and the denominator to their simplest form. Simplification helps in getting the expression into its most readable and useful format.
  • Combine the numerators while keeping the common denominator.
  • Look for opportunities to reduce any common factors from the numerator and the denominator.
  • Simplified fractions are easier to work with in subsequent algebraic operations.
In this solution, the combined expression \(\frac{3-5}{x-2}\) was simplified to \(\frac{-2}{x-2}\). It is important to carefully simplify to prevent errors in later calculations.
Algebraic Expressions
Algebraic expressions consist of variables, numbers, and operational signs (like addition and subtraction). They form the basis for solving equations and manipulating mathematical expressions.
  • Expressions can be simplified, added, subtracted, multiplied, or divided.
  • Understanding how to manipulate algebraic expressions is key to solving complex equations.
  • Identify like terms and be cautious with negative signs as they can change the meaning of an expression.
In rational expressions, like the one in our exercise, the operation on the expression hinges on understanding these algebraic principles. Rewriting and simplifying ensures accurate solutions while respecting the integrity of the expression.

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