Problem 46 - College Algebra

Question

Simplify the rational expression. $$ \frac{\frac{3}{x+1}+\frac{2}{x-1}}{\frac{x-1}{x+1}} $$

Step-by-Step Solution

Answer

\( \frac{5x - 1}{x-1} \)

Step 1: Combine the Numerator

First, combine the fractions in the numerator. We have the expression: \[\frac{3}{x+1} + \frac{2}{x-1}.\]To add these fractions, find a common denominator, which is \((x+1)(x-1)\), and rewrite each fraction:\[\frac{3(x-1)}{(x+1)(x-1)} + \frac{2(x+1)}{(x+1)(x-1)}.\]Now, combine these expressions into a single fraction:\[\frac{3(x-1) + 2(x+1)}{(x+1)(x-1)}.\]

Step 2: Simplify the Combined Expression

Next, expand and simplify the expression in the numerator:\[3(x-1) = 3x - 3,\]\[2(x+1) = 2x + 2.\]Add these together:\[3x - 3 + 2x + 2 = 5x - 1.\]This gives us the combined fraction:\[\frac{5x - 1}{(x+1)(x-1)}.\]

Step 3: Simplify the Whole Expression

Given the original complex fraction:\[\frac{\frac{5x - 1}{(x+1)(x-1)}}{\frac{x-1}{x+1}}.\]Dividing by a fraction is the same as multiplying by the reciprocal, so rewrite it as:\[\frac{5x - 1}{(x+1)(x-1)} \times \frac{x+1}{x-1}.\]Simplify this by canceling out the like terms:\[\frac{5x - 1}{x-1}.\]

Step 4: Verify if Further Simplification is Possible

Check if the resulting expression \( \frac{5x - 1}{x-1} \) can be further simplified. Since the numerator and denominator share no common factors, this is our final expression.

Key Concepts

Simplification Process

Simplifying a rational expression involves breaking down the expression into its simplest form. Here, we need to perform operations to reduce its complexity without changing its value. To begin, combining fractions is a key part of simplifying rational expressions. We start by finding a common denominator, which helps to add or subtract fractions conveniently.In our example:- Combine the fractions in the numerator: \( \frac{3}{x+1} + \frac{2}{x-1} \)- Find the common denominator: \( (x+1)(x-1) \)After rewriting each fraction with the common denominator, the next step involves expanding the terms in the numerals and then simplifying by combining like terms. After simplifying the numerator, the expression becomes more manageable and ready for further simplification, such as cancellation of common terms with the denominator. This meticulous process ensures that we arrive at the simplest possible form of the rational expression.

Fraction Operations

Operations with fractions, especially in rational expressions, often require addition, subtraction, multiplication, and division. Understanding these operations is crucial for simplifying expressions. **Adding Fractions** When adding fractions, finding a common denominator is critical: - This common base allows the fractions to merge into one single fraction by combining their numerators. - Convert each fraction to have the same denominator by multiplying relevant factors. **Multiplication of Fractions** Multiplication is straightforward compared to addition. Here, multiply the numerators and the denominators directly: - Use this principle when dealing with complex fractions or when simplifying expressions with shared terms. In our exercise, fraction division is converted into a multiplication operation: - By multiplying with the reciprocal, we simplify the complex fraction without altering its value. Mastering these operations allows for seamless simplification, transforming complex algebraic expressions into simple forms.

Algebraic Expressions

At the heart of rational expressions are algebraic expressions. They encompass variables, constants, and algebraic operations.In algebraic expression manipulation, especially in rational expressions:- Terms with variables like \( 3(x-1) \) and \( 2(x+1) \) are common and require expansion.**Expanding Expressions**- Distribute constants across terms within parentheses, simplifying to individual terms: \( 3(x-1) = 3x - 3 \). **Combining Like Terms**- Similar terms such as coefficients of \( x \) are summed up to form a coherent expression. These processes help take potentially complicated algebraic expressions to a simpler form, aiding in overall expression simplification. Finally, look for common terms that ensure the final expression is as simple as possible, free from redundancies or factorable elements. By mastering algebraic manipulations, rational expression simplification becomes much more intuitive.