Problem 40 - Prealgebra and Introductory Algebra

Question

Write the fractions in terms of the LCM of the denominators. $$\frac{x^{2}}{2 x-1}, \frac{x+1}{x+4}$$

Step-by-Step Solution

Answer

The fractions \(\frac{x^{2}}{2 x-1}\) and \(\frac{x+1}{x+4}\) when expressed in terms of the LCM of their denominators will be \( \frac{x^{2}*(x+4)}{(2 x-1)*(x+4)} \) and \( \frac{(x+1)*(2x-1)}{(x+4)*(2x-1)} \) respectively.

Step 1: Identify the denominators

The denominators of the fractions are \(2 x-1\) and \(x+4\).

Step 2: Calculate the Least Common Multiple (LCM)

Since the denominators are not like terms, the LCM is simply the product of these terms, that is, LCM = \((2x -1)*(x+4)\).

Step 3: Re-write the fractions

Express each fraction as an equivalent fraction that has \((2x-1)(x+4)\) as the denominator. This is done by multiplying numerator and denominator of each fraction by the factor it is missing from the LCM. For the first fraction: \( \frac{x^{2}}{2 x-1}=> \frac{x^{2}*(x+4)}{(2 x-1)*(x+4)} \). For the second fraction: \( \frac{x+1}{x+4} => \frac{(x+1)*(2x-1)}{(x+4)*(2x-1)} \).

Key Concepts

Understanding Fractions

Fractions are a crucial part of mathematics, representing parts of a whole. A fraction is written with two numbers, one above the other, separated by a line. The number on top is the numerator, indicating how many parts are considered. The bottom number is the denominator, showing the total number of equal parts the whole is divided into. In our exercise, fractions are used in algebraic expressions, such as \(\frac{x^2}{2x-1}\) and \(\frac{x+1}{x+4}\). Understanding how to manipulate fractions and make their denominators consistent is key in solving algebraic equations.

Role of Denominators

Denominators play a critical role in fractions. They define the division of the whole into equal parts. When working with multiple fractions, especially in algebraic contexts, having a common denominator simplifies calculations and operations such as addition or subtraction. To work with fractions that have different denominators, like in our example, they must be expressed in terms of a common denominator. This involves finding the least common multiple (LCM) of the denominators so all fractions can be rewritten to have the same base, simplifying the subsequent mathematical processes.

Applying Algebra with Fractions

Algebra introduces variables into mathematical expressions and equations, adding a layer of complexity to operations with fractions. Here, you replace numbers with letters or symbols (like \(x\)), and these variables follow specific algebraic rules. In algebra, when dealing with fractions such as \(\frac{x^2}{2x-1}\) and \(\frac{x+1}{x+4}\), it's essential to manipulate and simplify expressions to perform operations easily. This involves multiplying and factoring polynomials, allowing for direct comparisons and simplifications when the same variables and terms appear.

Performing LCM Calculations

The Least Common Multiple (LCM) is the smallest number or expression that is a multiple of each of the denominators in a set of fractions. When denominators include expressions rather than simple numbers, like \(2x-1\) and \(x+4\) in our example, determining the LCM involves algebraic multiplication.Here, the LCM is calculated by multiplying the different denominator expressions together: LCM = \((2x-1)(x+4)\). This product forms the new denominator for both fractions. Re-writing the fractions with this common denominator allows for straightforward addition, subtraction, or comparison operations.