Problem 59 - Algebra and Trigonometry

Question

\(55-64=\) Simplify the compound fractional expression. $$ \frac{\frac{5}{x-1}-\frac{2}{x+1}}{\frac{x}{x-1}+\frac{1}{x+1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3x+7}{x^2+2x-1} \).

Step 1: Find a Common Denominator for Each Fraction in the Numerator

First, look at the numerator \( \frac{5}{x-1} - \frac{2}{x+1} \). To subtract these fractions, you need a common denominator. The common denominator of \( x-1 \) and \( x+1 \) is \((x-1)(x+1)\). Rewriting each fraction with this common denominator, we have:\[ \frac{5(x+1)}{(x-1)(x+1)} - \frac{2(x-1)}{(x-1)(x+1)}.\]

Step 2: Compute the Numerator of the Expression

Now, perform the subtraction in the numerator:\[ \frac{5(x+1) - 2(x-1)}{(x-1)(x+1)}.\]Simplify the expression inside the numerator:\[ 5(x+1) = 5x + 5 \] and\[ 2(x-1) = 2x - 2. \]Substitute back into the expression:\[ 5x + 5 - 2x + 2 = 3x + 7. \]Thus the numerator simplifies to:\[ \frac{3x + 7}{(x-1)(x+1)}.\]

Step 3: Find a Common Denominator for Each Fraction in the Denominator

Now, look at the denominator \( \frac{x}{x-1} + \frac{1}{x+1} \). The common denominator for these fractions is also \((x-1)(x+1)\). Rewriting each fraction with this common denominator gives:\[ \frac{x(x+1)}{(x-1)(x+1)} + \frac{1(x-1)}{(x-1)(x+1)}.\]

Step 4: Compute the Denominator of the Expression

Combine the fractions in the denominator:\[ \frac{x(x+1) + 1(x-1)}{(x-1)(x+1)}.\]Simplify the expression in the numerator:\[ x(x+1) = x^2 + x \] and\[ 1(x-1) = x - 1. \]Add these together:\[ x^2 + x + x - 1 = x^2 + 2x - 1. \]Thus, the denominator simplifies to:\[ \frac{x^2 + 2x - 1}{(x-1)(x+1)}.\]

Step 5: Divide the Simplified Numerator by the Simplified Denominator

Dividing the simplified numerator by the simplified denominator gives:\[ \frac{\frac{3x+7}{(x-1)(x+1)}}{\frac{x^2+2x-1}{(x-1)(x+1)}}.\]The denominators \((x-1)(x+1)\) cancel out, leaving:\[ \frac{3x+7}{x^2+2x-1}.\]

Step 6: Provide the Final Answer

The original expression simplifies to:\[ \frac{3x+7}{x^2+2x-1}.\]

Key Concepts

Numerator and Denominator Operations

When dealing with compound fractions, we often encounter operations involving both the numerator and the denominator. Understanding how to manage these operations step by step is crucial for simplifying the expression.
The initial goal is to handle the fractions in the numerator and the denominator separately before moving on to the overall simplification.

• First, look at the fractions in the numerator. For example, in the expression \( \frac{5}{x-1} - \frac{2}{x+1} \), you need to perform subtraction, which requires finding a common denominator.
• Similarly, operations in the denominator, such as \( \frac{x}{x-1} + \frac{1}{x+1} \), require addition, again necessitating a common denominator.
• Each fraction's operations (subtraction for the numerator and addition for the denominator) should be handled individually before moving on to the simplification steps.

Taking these steps systematically ensures that you're prepared to tackle the next stages of simplifying compound fractions.

Common Denominators

To successfully subtract or add fractions, you need to find a common denominator. This process is essential in ensuring the fractions can be combined into a single expression.
For compound fractions, this step is doubly important as both the numerators and denominators might require it.

• In the example above, the common denominator for both \( \frac{5}{x-1} - \frac{2}{x+1} \) and \( \frac{x}{x-1} + \frac{1}{x+1} \) is \((x-1)(x+1)\).
• This common denominator allows each fraction to be rewritten as a fraction with the same base, making subtraction or addition possible.
• Once rewritten, the numerators can be combined using normal arithmetic operations, resulting in a single expression with the common denominator.

Remembering to find and use common denominators is a key skill in simplifying these types of expressions.

Algebraic Simplification

Algebraic simplification is the process of reducing complex expressions into their simplest forms. After obtaining common denominators and combining the fractions, you can focus on simplifying the algebraic expressions.
In our specific example, you see how both the numerator and the denominator simplify before tackling the compound fraction.

• The numerator \( \frac{5(x+1) - 2(x-1)}{(x-1)(x+1)} \) simplifies when you expand and combine like terms, resulting in \( 5x + 5 - 2x + 2 = 3x + 7 \).
• The denominator follows a similar path, producing an expression of \( x^2 + 2x - 1 \).
• Once both the numerator and denominator are simplified, you can divide them, simplifying further if possible, leading to \( \frac{3x+7}{x^2+2x-1} \).

Mastery of algebraic simplification techniques enables you to work through compound fractions efficiently and accurately.