Problem 48 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{3}{x+1}+\frac{1}{x-2}}{\frac{1}{x+1}-\frac{4}{x-2}} $$

Step-by-Step Solution

Answer

\( \frac{4x - 5}{-3x - 6} \)

Step 1 - Combine the numerators

First, find a common denominator for the terms in the numerator \(\frac{3}{x+1} + \frac{1}{x-2}\). The common denominator is \((x+1)(x-2)\). Rewriting both fractions, we get: \(\frac{3(x-2) + 1(x+1)}{(x+1)(x-2)} = \frac{3x - 6 + x + 1}{(x+1)(x-2)} = \frac{4x - 5}{(x+1)(x-2)}\).

Step 2 - Combine the denominators

Second, find a common denominator for the terms in the denominator \(\frac{1}{x+1} - \frac{4}{x-2}\). The common denominator is \((x+1)(x-2)\). Rewriting both fractions, we get: \(\frac{1(x-2) - 4(x+1)}{(x+1)(x-2)} = \frac{x-2 - 4x - 4}{(x+1)(x-2)} = \frac{-3x - 6}{(x+1)(x-2)}\).

Step 3 - Divide the fractions

Now, divide the combined numerator by the combined denominator: \(\frac{\frac{4x - 5}{(x+1)(x-2)}}{\frac{-3x - 6}{(x+1)(x-2)}}\). This simplifies to \(\frac{4x - 5}{-3x - 6}\).

Step 4 - Simplify the fraction

Simplify \(\frac{4x - 5}{-3x - 6}\). Since no common factors exist between the numerator and the denominator, the simplified form of the expression is \(\frac{4x - 5}{-3x - 6}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, a common denominator is essential. In the exercise, fractions like \(\frac{3}{x+1}\) and \(\frac{1}{x-2}\) need to be combined. The common denominator is the product of the individual denominators, which is \((x+1)(x-2)\). This allows us to rewrite each fraction.
For example, \(\frac{3}{x+1}\) becomes \(\frac{3(x-2)}{(x+1)(x-2)}\), and \(\frac{1}{x-2}\) becomes \(\frac{1(x+1)}{(x+1)(x-2)}\). Now, the fractions have a common denominator, making it possible to combine them easily.
Combining these fractions results in \(\frac{3(x-2) + 1(x+1)}{(x+1)(x-2)}\).
Utilizing the common denominator technique simplifies the initial step of the problem.

Numerators and Denominators

Numerators and denominators play distinct roles in fractions. The numerator is the top part, and the denominator is the bottom part of a fraction. To combine fractions, like in the exercise given, we adjusted both the numerators and denominators accordingly.
For the numerator, \( \frac{3}{x+1} + \frac{1}{x-2} \), here's what happens:

• Rewrite each fraction with the common denominator.
• Multiply the numerators by what's missing from the other fraction's denominator. For instance, 3 in the first fraction is multiplied by \( (x-2) \) and 1 by \( (x+1) \).

This creates the aligned numerators facilitating the addition. The same logic applies to adjusting the denominators in the example.

Fraction Division

Dividing fractions involves multiplying by the reciprocal. Having simplified both the numerator and denominator fractions, we need to divide them. For instance, our problem's result shifted to: \( \frac{\frac{4x-5}{(x+1)(x-2)}}{\frac{-3x-6}{(x+1)(x-2)}} \).
Division of these complex fractions simplifies to multiplication by flipping the second fraction: \( \frac{4x-5}{(x+1)(x-2)} \times \frac{(x+1)(x-2)}{-3x-6} \).
By cancelling out common terms, we directly simplify the resultant fraction: \( \frac{4x-5}{-3x-6} \). Thus, utilizing the reciprocal is vital in fraction division.

Fraction Simplification

Fraction simplification minimizes the expression to its simplest form. After adding and dividing the fractions in this exercise, we attain \( \frac{4x-5}{-3x-6} \).
Check for any common factors between the numerator (4x-5) and the denominator (-3x-6). If there were any, factorization and cancellation would be necessary.

• Analyze the numerator and denominator through basic factorization techniques.
• No common factors found indicate the fraction is already in its simplest form.

Therefore, the simplified form of the exercise's final expression remains as \( \frac{4x-5}{-3x-6} \). Fraction simplification is essential for clarity and reducing complexity.