Problem 67 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ \frac{x+\frac{1}{x+2}}{x-\frac{1}{x+2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{x^2 + 2x + 1}{x^2 + 2x - 1} \).

Step 1: Identify the Inner Fractions

In the given problem \( \frac{x + \frac{1}{x+2}}{x - \frac{1}{x+2}} \), notice that the inner fractions are \( \frac{1}{x+2} \). These terms need to be simplified within the larger expression.

Step 2: Find a Common Denominator

The common denominator for the fractions in the numerator and the denominator of the main fraction is \( x+2 \). Multiply both the terms in the numerator and the denominator by \( x+2 \) to eliminate the inner fractions.

Step 3: Simplify the Numerator

The numerator \( x + \frac{1}{x+2} \) becomes \( (x)(x+2) + 1 \) when multiplied by \( x+2 \). Simplifying this gives \( x^2 + 2x + 1 \).

Step 4: Simplify the Denominator

The denominator \( x - \frac{1}{x+2} \) becomes \( (x)(x+2) - 1 \) when multiplied by \( x+2 \). Simplifying this gives \( x^2 + 2x - 1 \).

Step 5: Write the New Expression

After multiplying through by the common denominator and simplifying, the original expression becomes \( \frac{x^2 + 2x + 1}{x^2 + 2x - 1} \).

Key Concepts

Compound Fractions

Compound fractions are also known as complex fractions. They are fractions where the numerator, the denominator, or both contain fractions themselves. This often looks like fractions stacked upon fractions.
You might see a compound fraction reading something like:

• \( \frac{a + \frac{b}{c}}{d + \frac{e}{f}} \)

To simplify, it's important to first evaluate the smaller, inner fractions. Once they are simplified or evaluated, the larger fractional structure can then be handled using common algebraic methods. Successfully managing these complex arrangements is fundamental in algebra, enabling you to simplify more complicated expressions effectively. Once grasped, it becomes straightforward to convert these into simpler fractional forms and solve them.

Simplifying Expressions

Simplifying expressions is a key step in dealing with algebraic problems, especially with compound fractions. The process often involves several sub-steps:

• Combining like terms
• Reducing fractions
• Using distributive properties

For compound fractions, simplifying involves eliminating the compound structure by multiplying through by a common denominator. In the example, multiplying both the numerator and denominator by the inner denominator, \(x+2\), helps turn the complex structure into a simpler one.
Once you accurately simplify the terms, resulting in an easier-to-read expression, you have laid the groundwork for solving or rewriting the entire equation. It is always vital to cross-check each simplification step to prevent mistakes, as algebra demands precision.

Common Denominator

Identifying a common denominator is essential when simplifying compound fractions or combining two fractional terms. A common denominator allows you to transform fractions with different denominators into comparable terms that can be easily added or subtracted. In reality, it’s about finding a mutual "language" for these fractions so they can be communicated together effectively.

For example, if you have fractions such as \( \frac{2}{3} \) and \( \frac{1}{4} \), their common denominator is 12. This allows you to write them as:

• \( \frac{8}{12} \)
• \( \frac{3}{12} \)

In our exercise, the common denominator for the inner fractions was \(x+2\). By multiplying the entire numerator and the denominator by this term, the fractions cancelled out appropriately. Consequently, this step streamlined our complex fraction into a simpler form, making it much easier to manage and solve. Remember, consistently finding and applying a common denominator will empower you for not just fractions, but algebraic problems of varying complexity.