Problem 68 - College Algebra with Modeling and Visualization

Question

Find the \(L C D\) for the rational expressions. $$ \frac{1}{2 x-1}, \frac{1}{x+1} $$

Step-by-Step Solution

Answer

The LCD is \((2x-1)(x+1)\).

Step 1: Identify Denominators

The given rational expressions have denominators of \(2x-1\) and \(x+1\). Identifying these denominators is the first step in finding the least common denominator (LCD).

Step 2: Determine the LCD

The least common denominator for rational expressions is the smallest expression that both denominators can divide into without leaving a remainder. In this case, the LCD must include both \(2x-1\) and \(x+1\) because they have no common factors.

Step 3: Write the LCD

Since both \(2x-1\) and \(x+1\) are needed, multiply them together to create the least common denominator: \((2x-1)(x+1)\).

Key Concepts

Least Common Denominator

Understanding the least common denominator (LCD) is crucial when dealing with rational expressions. The LCD is essentially the smallest expression that all denominators in a group can be divided into without any remainder.
To find the LCD, identify the individual denominators first. Let's illustrate this using the example \( \frac{1}{2x-1} \) and \( \frac{1}{x+1} \), where the denominators are \( 2x-1 \) and \( x+1 \), respectively.
Since these denominators have no common factors, the LCD is simply the product of these unique denominators, which results in \((2x-1)(x+1)\). By ensuring both rational expressions are rewritten with this common denominator, it becomes easier to perform operations like addition or subtraction on them.

Rational Expressions

A rational expression refers to a fraction where both the numerator and the denominator are polynomials.
These expressions behave similarly to regular fractions, but with polynomial expressions at play. Understanding how to manipulate rational expressions is important in algebra as it allows for operations like simplification and solving equations.

• Numerators and denominators are polynomials.
• Operations such as addition, subtraction, multiplication, and division often require the expressions to have the same denominator.

For instance, consider \( \frac{1}{2x-1} \) and \( \frac{1}{x+1} \). These are rational expressions because the numerators (1 in both cases) and denominators (\(2x-1\) and \(x+1\)) are polynomials. Making them share a common denominator simplifies their combination or comparison.

Denominators

Denominators are the expressions found at the bottom of a fraction and they play a vital role in algebra when working with rational expressions.
They dictate how fractions are combined and how their values compare.
For an expression like \( \frac{1}{2x-1} \), \( 2x-1 \) is the denominator.

• Each denominator contributes to calculating the least common denominator.
• A shared denominator allows for straightforward addition or subtraction.

In combining two expressions, a common denominator lets you align the fractions to easily add or subtract the numerators afterward. This is why finding the LCD is so important: it paves the way for seamless fraction work and simplifies what could otherwise be a complex task.