Problem 114 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Problem

We are given an expression \( \frac{\frac{1}{2} - x}{\frac{1}{x} - 2} \) and we need to simplify it by performing algebraic manipulations.

Step 2: Simplify the Numator

Focus first on the numerator \( \frac{1}{2} - x \). There's no further simplification possible here without using specific values, so we'll maintain it as it is for now.

Step 3: Simplify the Denominator

Turn our attention to the denominator \( \frac{1}{x} - 2 \). We want to combine these into a single fraction: \( \frac{1}{x} - \frac{2x}{x} = \frac{1 - 2x}{x} \).

Step 4: Combine into a Single Fraction

Now, replace the denominator in the main expression: \( \frac{\frac{1}{2} - x}{\frac{1 - 2x}{x}} \). We can rewrite the entire expression as \( \left( \frac{1}{2} - x \right) \times \left( \frac{x}{1 - 2x} \right) \).

Step 5: Simplify the Expression

Distribute \(x\) in the numerator: \( \frac{x \left( \frac{1}{2} - x \right) }{1 - 2x} = \frac{\frac{x}{2} - x^2}{1 - 2x} \). Now the expression is in its simplest form since no terms cancel out.

Key Concepts

Rational Expressions

Rational expressions are like fractions but with polynomials in their numerators and denominators. Much like fractions, these expressions involve division. To operate with rational expressions, similar principles used in fractional operations apply.

Rational expressions can often be simplified similarly to numerical fractions. This involves canceling common factors in the numerator and denominator. However, care must be taken since polynomial expressions can have variables, which must be taken into account to avoid incorrectly simplifying away terms that change the expression's validity.

• When simplifying, look for common factors.
• Check for terms that could potentially be zero to avoid division by zero.

In our example expression, \[ \frac{\frac{1}{2} - x}{\frac{1}{x} - 2} \], we deal with combining rational expressions within each part—the numerator and the denominator. This process treats each part as a rational expression and follows simplification rules until the expression is fully simplified.

Fractions in Algebra

Fractions in algebra are expressions that hold variables in either their numerators, denominators, or both. The operation on these fractions follows similar rules to those applied when dealing with numerical fractions but with an additional emphasis on handling variables carefully.

In fractional algebraic expressions, understanding how to manipulate and simplify fractions is essential. It involves:

• Combining fractions by finding common denominators.
• Rewriting fractions to allow for the simplification.
• Transforming complex fractions into simpler forms.

In our given exercise, \[ \frac{\frac{1}{2} - x}{\frac{1}{x} - 2} \], the denominator is dealt with by finding a common denominator to manifest a single fraction: \[ \frac{1 - 2x}{x} \]. This step is crucial in turning the entire expression into a set that can be easily simplified overall.

Numerator and Denominator Manipulation

Manipulating the numerator and the denominator separately is a critical step in simplifying algebraic fractions. Each part of the fraction can be simplified separately, sometimes allowing the entire expression to become much more manageable.

Key techniques include:

• Rewriting numerators or denominators to expose common factors.
• Combining like terms to decrease the complexity.
• Using distributive properties to reshape into simple forms.

In our exercise, the original numerator \( \frac{1}{2} - x \) remained as-is since it was already in its simplest form. However, the denominator \( \frac{1}{x} - 2 \) transformed into \( \frac{1 - 2x}{x} \), facilitating the conversion of our full expression into \[ \left( \frac{1}{2} - x \right) \times \left( \frac{x}{1 - 2x} \right) \]. After cross-multiplying, it finally simplifies to the elegant form \[ \frac{\frac{x}{2} - x^2}{1 - 2x} \]. This illustrates how targeting the numerator and denominator individually empowers a more effective overall simplification.