Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expression. \(\frac{\frac{3}{x+1}+\frac{2}{x-1}}{\frac{x-1}{x+1}}\)

Step-by-Step Solution

Answer

The simplified expression is \(5x - 1\).

Step 1: Identify the Expression

The given expression is a complex rational expression:\[\frac{\frac{3}{x+1} + \frac{2}{x-1}}{\frac{x-1}{x+1}}\].It consists of two fractions in the numerator and one fraction in the denominator.

Step 2: Simplify the Numerator

To simplify the numerator, \(\frac{3}{x+1} + \frac{2}{x-1}\), find a common denominator. The common denominator is \((x+1)(x-1)\).Rewriting each fraction with the common denominator:\[\frac{3}{x+1} = \frac{3(x-1)}{(x+1)(x-1)},\quad \frac{2}{x-1} = \frac{2(x+1)}{(x+1)(x-1)}\]Add these fractions:\[\frac{3(x-1) + 2(x+1)}{(x+1)(x-1)}\]Combine like terms:\[\frac{3x - 3 + 2x + 2}{(x+1)(x-1)} = \frac{5x - 1}{(x+1)(x-1)}\].

Step 3: Simplify the Denominator

The denominator of the entire expression is \(\frac{x-1}{x+1}\). This part is already simplified.

Step 4: Division of Fractions

Divide the simplified numerator by the denominator. This involves multiplying by the reciprocal:\[\frac{\frac{5x-1}{(x+1)(x-1)}}{\frac{x-1}{x+1}} = \left(\frac{5x-1}{(x+1)(x-1)}\right) \times \left(\frac{x+1}{x-1}\right)\]The \((x+1)(x-1)\) part in the denominator cancels with corresponding terms:\[= 5x - 1\].

Step 5: Final Simplification

After canceling, the expression simplifies to \(5x - 1\).There are no further terms to simplify.

Key Concepts

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both are themselves fractions. These are often seen in algebra problems and might look daunting at first. However, they follow the same principles as simpler fractions.
An example of a complex fraction is: \[ \frac{\frac{3}{x+1} + \frac{2}{x-1}}{\frac{x-1}{x+1}} \]To simplify a complex fraction, it's helpful to tackle one part at a time. By breaking down the expression, we focus first on simplifying the numerator and then the denominator before dealing with division. This methodical approach eases the complexity and helps avoid overwhelming steps in tackling the problem. Remember, solving is like untangling a puzzle — one piece at a time.

Simplification

The process of simplification in rational expressions involves reducing the expression to its simplest form. This often involves removing redundancies and combining like terms. By simplifying, you can make complex expressions easier to handle.
Let's take for example the numerator in our problem:\[ \frac{3}{x+1} + \frac{2}{x-1} \]To simplify, we first find a common denominator, which in this case is \((x+1)(x-1)\). With a common denominator, combine the terms:

• Convert \(\frac{3}{x+1}\) to \(\frac{3(x-1)}{(x+1)(x-1)}\)
• Convert \(\frac{2}{x-1}\) to \(\frac{2(x+1)}{(x+1)(x-1)}\)
• Add the fractions to get \(\frac{5x - 1}{(x+1)(x-1)}\)

This approach makes it easier to manage the expression as a whole.

Common Denominator

Finding a common denominator is a crucial step in simplifying complex fractions. It allows you to combine fractions by ensuring they are set over the same base, so to speak. In rational expressions like our example, it's vital to efficiently manage multiple terms under one roof.
Consider the fractions in the numerator:

• \(\frac{3}{x+1}\)
• \(\frac{2}{x-1}\)

The common denominator is the product of the unique terms in these denominators. Thus, we obtain:\[ (x+1)(x-1) \]Rewriting each part using this common base lets you combine the expressions in the numerator into one. This makes the expression easier to manage and eventually simplifies the entire complex fraction.

Algebraic Expressions

Algebraic expressions involve variables, numbers, and operations. These expressions can become complex when dealing with fractions, yet understanding their structure makes simplification easier.
An algebraic expression could include operations such as addition, subtraction, along with multiplications involving variables like \(x\). A clear grasp of how these operations interact within the expressions is necessary for simplification.In our exercise, algebraic manipulation like distributing and combining terms is used:

• \(3(x-1)\) becomes \(3x - 3\)
• \(2(x+1)\) becomes \(2x + 2\)
• Combining the above gives \((3x - 3) + (2x + 2) = 5x - 1\)

This breakdown into manageable steps not only demystifies the concept but also forms a foundation for working with more advanced algebraic expressions.