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Chapter 11: Problem 80

Simplify. $$\frac{x}{x-4}+\frac{5}{x+5}-\frac{11 x-8}{x^{2}+x-20}$$

Short Answer

Expert verified
The simplified form of the given expression is \(\frac{4x+40}{x^2+x-20}\).

Step by step solution

01

Identify Common Denominator

The common denominator for all the fractions is given by the product of individual denominators, here it is \( (x-4)(x+5) \).
02

Adjusting the Fractions

Express each fraction with the common denominator, giving us: \[ \frac{x(x+5)}{(x-4)(x+5)} + \frac{5(x-4)}{(x+5)(x-4)} - \frac{11x-8}{x^2+x-20} \]
03

Combine the Fractions

Now combine all the fractions by adding or subtracting the numerators: \[ \frac{x(x+5)+5(x-4)-(11x-8)}{x^2+x-20} \]
04

Simplify the Expression

Simplify the numerator and denominator to get the final answer. The simplified form is \[ \frac{4x+40}{x^2+x-20} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
In the world of fractions, the common denominator is the key to unlocking the simplest form of an expression. When faced with multiple fractions, finding a common denominator is a crucial step because it allows all the fractions to be combined into a single fraction, making the equation simpler to handle.

Think of the common denominator as a common language that all fractions in an equation need to speak in order to work together. In mathematical terms, it is the least common multiple of the denominators. In the exercise provided, the common denominator for the fractions \(\frac{x}{x-4}\), \(\frac{5}{x+5}\), and \(\frac{11x-8}{x^2+x-20}\) is the product of the unique factors found in their denominators, which in this case is \( (x-4)(x+5) \).

Once determined, each fraction is adjusted, or \

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