Problem 8 - Introductory Algebra for College Students

Question

Simplify complex rational expression by the method of your choice. \(\frac{\frac{2}{3}-x}{\frac{2}{3}+x}\)

Step-by-Step Solution

Answer

The simplified expression of \(\frac{2}{3}-x}{\frac{2}{3}+x}\) is \(\frac{2-x}{2+x}\)

Step 1: Rewriting the Problem

To simplify a complex rational function, start by getting rid of the fractions in the numerator and denominator. To do this, find a common denominator for the fractions, which in this case, is \(3x\). Contingent on this, we rewrite the expression as follows: \(\frac{2(3x) - x(3x)}{2(3x) + x(3x)}\)

Step 2: Simplify the Expression

The next step is to simplify the expressions in the numerator and denominator: \(\frac{6x-3x^{2}}{6x+3x^{2}}\)

Step 3: Factorize the Expression

In order to simplify further, factorize the expressions in the numerator and the denominator by the greatest common divisor, which is \(3x\): \(\frac{3x(2-x)}{3x(2+x)}\)

Step 4: Cancel out Common Factors

Now cancel out the common factor of \(3x\) in both numerator and denominator to get the simplified fraction: \(\frac{2-x}{2+x}\)

Key Concepts

Complex Fractions

Complex fractions might sound intimidating, but they are essentially fractions within fractions. Recognizing this is the first step in simplifying them. We start with a number or expression in the numerator and another in the denominator, each of which could also be a fraction. Here's what to keep in mind:

• The goal is to combine these into a single rational expression without fractions.
• This involves algebraic procedures to simplify both the numerator and the denominator.

To simplify complex fractions like \( \frac{\frac{2}{3}-x}{\frac{2}{3}+x} \), it’s essential to get rid of inner fractions. This can typically be done by identifying a common denominator for the fractions within the numerator and denominator. Once a common denominator is established, we rewrite the expressions to eliminate these fractions, making further simplification possible.

Factoring Expressions

Factoring is like finding the building blocks of an expression. When simplifying rational expressions, factoring can break down more complex parts into simpler ones, which can then be combined or canceled. Here’s how it works:

• Identify any common factors within the expressions involved.
• Express these common factors as a product.

In our expression \( \frac{6x-3x^{2}}{6x+3x^{2}} \), both the numerator and the denominator can be factored by extracting the greatest common factor, which is \(3x\). This yields \( \frac{3x(2-x)}{3x(2+x)} \), a pivotal step that prepares us for cancellation, and thus, further simplification.

Algebraic Simplification

Algebraic simplification is the process of making expressions as simple as possible. Once we have factored the expression, the next step is to simplify it by removing unnecessary components. The focus here is on:

• Reducing expressions to their simplest form.
• Canceling out identical terms in the numerator and denominator.

In \( \frac{3x(2-x)}{3x(2+x)} \), simplification is achieved by canceling the \(3x\) terms in both the numerator and the denominator. This simplifies our expression to \( \frac{2-x}{2+x} \). Always ensure that what you’re canceling is truly a duplicate in both the numerator and the denominator. This can prevent errors.

Common Denominator

Finding a common denominator is crucial when dealing with complex fractions. It’s the groundwork that allows the fraction to be simplified effectively. Here's the step-by-step guide:

• Identify all fractions involved in your expression.
• Determine the common denominator for these fractions.

In the case of \( \frac{\frac{2}{3}-x}{\frac{2}{3}+x} \), the denominator of 3 for the fractions suggests \(3x\) as a common multiple due to the presence of \(x\) terms. Using this common denominator streamlines the transformation of the initial complex fraction into a more straightforward format like \( \frac{6x-3x^{2}}{6x+3x^{2}} \), which can then be factored and simplified further. Mastering the finding of common denominators is a foundational skill for simplifying complex algebraic fractions.