Problem 40 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{10}{3 n-4}-\frac{5}{4-3 n}\)

Step-by-Step Solution

Answer

\( \frac{15}{3n-4} \).

Step 1: Simplify the denominators

Note that \( 4 - 3n = -(3n - 4) \). Therefore:
\( \frac{5}{4-3n} = \frac{5}{-(3n-4)} = -\frac{5}{3n-4} \).

Step 2: Combine

\( \frac{10}{3n-4} - \left(-\frac{5}{3n-4}\right) = \frac{10}{3n-4} + \frac{5}{3n-4} = \frac{15}{3n-4} \).

Key Concepts

Rational Expressions

When we talk about rational expressions, we're essentially dealing with fractions that have polynomials in the numerator, the denominator, or both. Just like regular fractions, rational expressions follow the same rules for operations and simplifications. These expressions are a pivotal part of algebra because they often simplify complex problems into more manageable forms.

The given exercise involves the operation of subtraction between two rational expressions. Here, each rational expression represents a fraction with different polynomial denominators. Understanding this concept is crucial because rational expressions can behave differently from simple numerical fractions due to their variable components.

• Rational Expressions are similar to regular fractions but include variables.
• Operations such as addition, subtraction, multiplication, and division can be performed on them.
• Care must be taken to ensure denominators are not zero, which would make the expression undefined.

In the given expressions, we're subtracting fractions with the denominators of \(3n - 4\) and \(4 - 3n\). Recognizing and rewriting these denominators correctly is key to simplifying them.

Simplifying Fractions

Simplification is a critical step when working with fractions or rational expressions. The goal is to reduce the expression to its most basic form without changing its value. In rational expressions, this involves canceling common factors between the numerator and the denominator.

To simplify the given expression, first, observe if the terms in the denominators can be related. Notice that \(3n - 4\) and \(4 - 3n\) are actually negatives of each other. This observation becomes crucial for their simplification. If we adjust one denominator, by factoring out a -1 from \(4-3n\), we can make them identical: \(4 - 3n = -(3n - 4)\). This trick helps us in simplifying our operation.

• Rewriting similar denominators is essential as it allows subtraction or addition to proceed.
• When common factors appear in the numerator and denominator, they can be canceled out.
• This often leads to a simplified expression with easier calculations.

Through clever manipulation of factors, simplification becomes a shorter path to finding the final result.

Denominator Analysis

Denominator analysis is a vital skill when dealing with rational expressions. This step involves understanding and manipulating the denominators to perform operations efficiently. Since denominators control the 'bottom' part of a fraction's structure, knowing how to manage them is essential.

In our exercise, we begin by comparing the denominators \(3n - 4\) and \(4 - 3n\). They seem alike, but a sign difference exists. By factoring out a -1, we can make them the same. This step ensures that we can correctly combine the fractions. The key steps in denominator analysis include:

• Identifying patterns or similarities in denominators that can be exploited.
• Using algebraic tricks like factoring or distributing to align denominators.
• Ensuring that the operation does not make any denominator equal to zero, as that would break the expression.

Mastering denominator analysis leads to smooth transitions between different stages of rational expression operations and ensures accurate results.