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Chapter 7: Problem 41

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

Short Answer

Expert verified
The simplified form of the complex rational expression is \(\frac{5x+33}{x^{2}+9x+28}\)

Step by step solution

01

Identify the Least Common Denominator (LCD)

Look at the denominator of each mini-fraction and identify what they have in common. With \(x^{2}+2 x-15\), \(x-3\), and \(x+5\), we find that the LCD is \(x^{2}+2 x-15 = (x+5)(x-3)\).
02

Multiply Through by the LCD

Now, to simplify the expression, multiply the numerator and denominator of the complex fraction by the LCD. This operation is \( (x+5)(x-3) \times \frac{\frac{6}{x^{2}+2 x-15}-\frac{1}{x-3}}{\frac{1}{x+5}+1}\).
03

Simplify the Numerator

Multiplying through the numerator we get \(6(x+5)-1(x-3)\). Which simplifies to \(6x+30-x+3 = 5x+33\).
04

Simplify the Denominator

Multiplying through the denominator yields \((x+5)(x+5)-(x-3) = x^{2}+10x+25-x+3 = x^{2}+9x+28\).
05

Write the Simplified Fraction

The complex fraction simplifies to \(\frac{5x+33}{x^{2}+9x+28}\). This cannot be simplified further, so it is the solution.

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