Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expression. \(\frac{\frac{2 x}{3}+\frac{4 x}{7}}{\frac{x}{2}}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{52}{21}\).

Step 1: Simplify the Numerator

To simplify the numerator, find a common denominator for the fractions \(\frac{2x}{3}\) and \(\frac{4x}{7}\). The common denominator is 21. Rewrite each fraction with this common denominator: \(\frac{2x}{3} = \frac{14x}{21}\) and \(\frac{4x}{7} = \frac{12x}{21}\). Add these to get \(\frac{26x}{21}\).

Step 2: Rewrite the Expression

Replace the original numerator in the expression with the simplified form: \(\frac{\frac{26x}{21}}{\frac{x}{2}}\).

Step 3: Simplify by Dividing Fractions

To simplify \(\frac{\frac{26x}{21}}{\frac{x}{2}}\), multiply by the reciprocal of \(\frac{x}{2}\). So it becomes \(\frac{26x}{21} \times \frac{2}{x}\).

Step 4: Perform Multiplication

Multiply the numerators: \(26x \times 2 = 52x\). Multiply the denominators: \(21 \times x = 21x\). This gives \(\frac{52x}{21x}\).

Step 5: Cancel Common Factors

Cancel out \(x\) in the numerator and denominator: \(\frac{52x}{21x} = \frac{52}{21}\). This is the simplified form.

Key Concepts

Simplification

Simplification of rational expressions involves reducing them to their simplest form. A rational expression is made up of a ratio of two polynomials. To simplify such expressions efficiently, identify and eliminate any common factors from the numerator and the denominator. This process involves:

• Combining like terms where similar variables and powers are present
• Performing arithmetic operations

The goal is to make the expression much simpler and easier to analyze. In the given solution, simplifying the rational expression ultimately boils down to canceling the common factor of \(x\) from both the numerator and denominator, reducing \(\frac{52x}{21x}\) to \(\frac{52}{21}\).
Simplification makes further algebraic manipulation more manageable.

Common Denominators

Finding a common denominator is crucial when adding or subtracting fractions, as it allows these operations to be performed on a consistent basis. A common denominator is essentially a shared multiple of the denominators involved. For the fractions \(\frac{2x}{3}\) and \(\frac{4x}{7}\), the least common denominator is 21.

• Rewrite each fraction so that they have this common denominator
• Ensure that the equivalent expressions retain the same value as the original fractions
• This adjustment allows for straightforward addition or subtraction of the numerators

By finding a common denominator, complex expressions in both mathematical computations and real-world applications become more navigable.

Fraction Operations

Performing operations with fractions—be it addition, subtraction, multiplication, or division—follows specific rules. For expressions like \(\frac{\frac{26x}{21}}{\frac{x}{2}}\), understanding these operations becomes key.

• Addition/Subtraction: Only possible with common denominators
• Multiplication: Multiply numerators together and denominators together
• Division: Multiply by the reciprocal of the divisor

For the problem at hand, the solution illustrates dividing fractions by converting division into multiplication using the reciprocal. This operation led to the simplification of the expression from complex fractions into a much simpler form.

Reciprocal Multiplication

Reciprocal multiplication transforms division problems into multiplication ones, aiding in simplifying rational expressions. To find the reciprocal of a fraction, swap its numerator and denominator. For \(\frac{x}{2}\), the reciprocal is \(\frac{2}{x}\).
By multiplying \(\frac{26x}{21}\) by the reciprocal \(\frac{2}{x}\), you effectively handle operations that require division. This step is essential since multiplying instead of dividing is generally more intuitive and aligns with the rules of fraction operations:

• Transforms complex division into straightforward multiplication
• Allows simplification by cancellation of similar terms

Reciprocal multiplication streamlines the problem-solving process in rational expressions and enhances computational efficiency.