Problem 38 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{2}{x}+\frac{x}{2}}{\frac{2}{x}-\frac{x}{2}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{4+x^2}{4-x^2} \).

Step 1: Understand the Problem

We have a complex fraction: \( \frac{\frac{2}{x}+\frac{x}{2}}{\frac{2}{x}-\frac{x}{2}} \). The goal is to simplify it by combining the numerators and denominators into simpler expressions.

Step 2: Find a Common Denominator for Numerators

To simplify the complex fraction, start with the numerator \( \frac{2}{x} + \frac{x}{2} \). The least common denominator (LCD) for these fractions is 2x. Rewrite each fraction with the common denominator:\[ \frac{2}{x} = \frac{2 \cdot 2}{2x} = \frac{4}{2x}, \quad \frac{x}{2} = \frac{x \cdot x}{2x} = \frac{x^2}{2x} \] Thus, the sum becomes:\[ \frac{4}{2x} + \frac{x^2}{2x} = \frac{4 + x^2}{2x} \]

Step 3: Find a Common Denominator for Denominators

Repeat the process for the denominator \( \frac{2}{x} - \frac{x}{2} \). Again use 2x as the common denominator:\[ \frac{2}{x} = \frac{4}{2x}, \quad \frac{x}{2} = \frac{x^2}{2x} \] Thus, the difference becomes:\[ \frac{4}{2x} - \frac{x^2}{2x} = \frac{4 - x^2}{2x} \]

Step 4: Form the New Fraction

Now, substitute the simplified forms back into the original complex fraction:\[ \frac{\frac{4 + x^2}{2x}}{\frac{4 - x^2}{2x}} \]

Step 5: Simplify by Division

Apply the rule for dividing fractions, \( \frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c} \):\[ \frac{4+x^2}{2x} \times \frac{2x}{4-x^2} \] Cancel the common terms, which are \(2x\):\[ = \frac{4+x^2}{4-x^2} \]

Step 6: Final Simplified Expression

The expression \( \frac{4+x^2}{4-x^2} \) is already in its simplest form as there are no common factors in the numerator and denominator.

Key Concepts

Simplifying Expressions

Simplifying expressions is about making them easier to read and understand, without changing their value. For complex fractions, this involves combining fractions within a fraction.
This process often requires finding a way to combine terms that have different denominators.

• Start by addressing the parts separately. Focus on the numerator first, then the denominator.
• Look for a common denominator to rewrite the terms so they share the same base.
• Combine terms by adding or subtracting their numerators.
• Repeat this process for the denominator before putting it all together.
• Simplification might involve reducing fractions or eliminating shared factors.

Ultimately, a simplified expression will have fewer terms and be more straightforward to work with. The goal is a clear, concise expression that's easier to evaluate or manipulate in further math work.

Least Common Denominator

Finding the least common denominator (LCD) is a pivotal step in working with fractions, especially when adding or subtracting them. It is the smallest number that all of the denominators divide into evenly.
When simplifying complex fractions, determining the LCD is vital to combine terms effectively.

• Inspect the denominators of each term you need to combine.
• Identify a common multiple that allows each fraction to be expressed with the same denominator.
• Rewrite each fraction with this LCD by adjusting their numerators accordingly.
• The LCD simplifies the addition or subtraction, allowing you to merge terms seamlessly.

This approach enables you to work with a single unified denominator, making the rest of the process much more manageable.

Algebraic Fractions

Algebraic fractions are expressions that contain fractions with variables in the numerator, denominator, or both. These fractions introduce additional steps in simplification due to the presence of variables.

• Handling algebraic fractions requires careful manipulation of both numbers and variables.
• The process often involves factoring expressions to identify and cancel out common factors.
• When simplifying, it’s essential to look for any opportunity to reduce terms by cancelling variables that appear both in the numerator and denominator.
• Always consider domain restrictions, as variables can introduce restrictions on values that keep the denominator from being zero.

Understanding and simplifying algebraic fractions refine problem-solving skills and enhance the ability to navigate more complex algebraic tasks.