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Chapter 0: Problem 71

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 complex rational expression is \(\frac{1-x}{x-3}\)

Step by step solution

01

Factor the Quadratic Expression

The quadratic expression in the denominator of the first fraction is \(x^{2} + 2x - 15\). This can be factored as \((x + 5)(x - 3)\), by finding two numbers that multiply to -15 and add to 2.
02

Simplify the Numerator

The numerator has two terms, \(\frac{6}{x^{2}+2 x-15}\) and \(-\frac{1}{x-3}\). Since we've factored \(x^{2}+2 x-15\) as \((x + 5)(x - 3)\), the first term becomes \(\frac{6}{(x + 5)(x - 3)}\). Now that the denominators of the first and second term are the same, we can subtract the second term from the first, to get \(\frac{6-1(x + 5)}{(x-3)(x+5)}\) = \(\frac{1-x}{(x-3)(x+5)}\).
03

Simplify the Denominator

The denominator is \(\frac{1}{x+5} + 1\). The number 1 can be written as \(\frac{x+5}{x+5}\), so that the denominator becomes \(\frac{1 + (x + 5)}{x + 5}\) = \(\frac{x + 6}{x + 5}\)
04

Divide Numerator by Denominator

The entire expression is a division of the numerator by the denominator, which is equivalent to the numerator multiplied by the reciprocal of the denominator. Thus, the expression becomes \(\frac{1-x}{(x-3)(x+5)} * \frac{x+5}{x+6}\). This simplifies to \(\frac{(1-x)(x+5)}{(x-3)(x+6)}\)
05

Cancel out common factors

Finally, the factor \(x+5\) in the numerator can be canceled with the same factor in the denominator, simplifying the expression to \(\frac{1-x}{x-3}\). Therefore, the simplified rational expression is \(\frac{1-x}{x-3}\)

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

If you are given a quadratic equation, how do you determine which method to use to solve it?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3,00\) to produce cach package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).
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