Problem 50 - Elementary Algebra

Question

For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{3}{x^{2}}-\frac{2}{x}}{\frac{4}{x}-\frac{7}{x^{2}}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{3-2x}{4x-7}\).

Step 1: Rewrite the Complex Fraction

The given complex fraction is \(\frac{\frac{3}{x^{2}}-\frac{2}{x}}{\frac{4}{x}-\frac{7}{x^{2}}}\). Rewrite it as a division of two fractions: \( \frac{3}{x^2} - \frac{2}{x} \) divided by \( \frac{4}{x} - \frac{7}{x^2} \).

Step 2: Identify Common Denominators

Identify the common denominators of the fractions in both the numerator and denominator. The common denominator for both the numerator and denominator is \(x^2\).

Step 3: Adjust the Numerator

Rewrite the numerator so it has the common denominator \(x^2\):\[ \frac{3}{x^2} - \frac{2}{x} = \frac{3}{x^2} - \frac{2 \cdot x}{x^2} = \frac{3 - 2x}{x^2} \]

Step 4: Adjust the Denominator

Rewrite the denominator so it has the common denominator \(x^2\):\[ \frac{4}{x} - \frac{7}{x^2} = \frac{4 \cdot x}{x^2} - \frac{7}{x^2} = \frac{4x - 7}{x^2} \]

Step 5: Simplify the Overall Fraction

Now divide the simplified numerators and denominators:\[ \frac{\frac{3-2x}{x^2}}{\frac{4x-7}{x^2}} = \frac{3-2x}{x^2} \times \frac{x^2}{4x-7} = \frac{3-2x}{4x-7} \]The \(x^2\) cancels out, leaving us with \(\frac{3-2x}{4x-7}\).

Step 6: Verify Final Simplification

Ensure that the result cannot be simplified further. Verify that there are no common factors in the final expression \(\frac{3-2x}{4x-7}\). The expression is fully simplified.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. They allow us to represent real-world relationships in a mathematical form and often appear in equations and functions.
Variables, mostly denoted by letters, such as \( x \) in our problem, stand for unknown values that we aim to find or simplify. When working with expressions involving variables, we assume that these variables can take on different values, changing the expression's value accordingly.
In the provided exercise, both the numerator and the denominator are algebraic expressions. They consist of fractions that include variables as part of their denominators. Understanding and manipulating these expressions requires knowledge of basic operations and properties such as the distributive property and how to combine like terms effectively.

Fraction Simplification

Fraction simplification is the process of making a fraction as simple as possible. This means removing any common factors between the numerator and the denominator. For complex fractions, like in our exercise, this involves multiple steps.
Here's how you simplify:

• Rewrite Complex Fractions: Start by expressing the top and bottom parts as single fractions. We rewrote \(\frac{\frac{3}{x^2}-\frac{2}{x}}{\frac{4}{x}-\frac{7}{x^2}}\) as \( \frac{3-2x}{x^2} : \frac{4x-7}{x^2} \).
• Find a Common Denominator: Recognize the greatest common factor for each part. In this case, \(x^2\) is used to standardize both the numerator and the denominator.
• Combine and Simplify: Multiply each part by the common denominator to allow for the cancellation of terms. Simplifying involves cancelling out any factors that both the numerator and denominator share.

By simplifying, we ensure that fractions are easier to work with and understand in further calculations or problem-solving steps.

Mathematical Operations

Mathematical operations, such as addition, subtraction, multiplication, and division, are everything in simplifying algebraic expressions. Each operation follows specific rules, especially when fractions and variables are involved.
1. **Addition and Subtraction on Fractions**: Only combine fractions when they have the same denominator. If not, find a common denominator as seen in the problem, where \(x^2\) was used to uniform both fractions in the numerator and denominator.
2. **Multiplication**: Multiply across the numerators and denominators when handling fractions. This process was used in simplifying the final expression in the exercise by cancelling \(x^2\), leaving us with only \(\frac{3-2x}{4x-7}\).
3. **Division**: This is performed by multiplying by the reciprocal. In the given problem, dividing \(\frac{3-2x}{x^2}\) by \(\frac{4x-7}{x^2}\) was resolved by multiplying with the reciprocal of \(\frac{4x-7}{x^2}\).
Understanding these operations helps in breaking down complex problems into simpler parts, making it easier to manage algebraic expressions and fractions.