Problem 70 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{\frac{6}{x^{2}+2 x-15}-\frac{1}{x-3}}{\frac{1}{x+5}+1}$$

Step-by-Step Solution

Answer

The simplified form of the given complex rational expression is \( \frac{9-x}{x^2-10} \).

Step 1: Identify the Denominator

First, let's identify the denominators that will be used for obtaining a common denominator. They are \(x^2+2x-15\), \(x-3\), \(x+5\), and \(1\). Note that a fraction that appears not to have a denominator actually has a denominator of 1.

Step 2: Factorize the Quadratic Expression

Factorize \(x^2+2x-15\) to \( (x-3)(x+5) \). This will allow us to easily identify the least common denominator (LCD) among the denominators.

Step 3: Obtain the Least Common Denominator (LCD)

Based on the factorization, the LCD is \( (x-3)(x+5) \).

Step 4: Simplify the Complex Fraction

Write each fraction in the complex fraction in terms of the LCD, and simplify. The complex fraction becomes, \( \frac{6 - (x-3)}{(x+5) + (x-3)(x+5)} \). Then further simplification yields, \( \frac{6 - x + 3}{x + 5 + x^2 - 15} \). Then simplify to, \( \frac{9-x}{x^2-10} \).

Step 5: Reduce to Lowest Terms

Here, unfortunately, the complex fraction cannot be further reduced since there are no common factors in the numerator and the denominator, other than 1. Thus, \( \frac{9-x}{x^2-10} \) is the simplified form of the original complex fraction.