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Chapter 7: Problem 85

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{5}{x^{2}-25}+\frac{4}{x^{2}-11 x+30}-\frac{3}{x^{2}-x-30}$$

Short Answer

Expert verified
\(\frac{6x + 5}{(x-5)(x+5)(x-6)}\)

Step by step solution

01

Factor the Denominators

Start by factoring the denominator of each fraction. The denominators \(x^{2}-25\), \(x^{2}-11x+30\), and \(x^{2}-x-30\) can be factored following the rules of factoring quadratic polynomials. Factored forms are \( (x-5)(x+5) \), \( (x-6)(x-5) \) and \( (x-6)(x+5) \) respectively.
02

Determine the Least Common Denominator (LCD)

The least common denominator should be the one that can be divided evenly by each of the denominators. Observing the factored forms, it's clear that the LCD is \( (x-5)(x+5)(x-6) \).
03

Convert Each Fraction to Have the LCD as the Denominator

To convert each fraction, multiply the top and bottom by what's missing from the LCD. So, \(\frac{5}{(x-5)(x+5)}\) becomes \(\frac{5(x-6)}{(x-5)(x+5)(x-6)}\), \(\frac{4}{(x-5)(x-6)}\) becomes \(\frac{4(x+5)}{(x-5)(x+5)(x-6)}\), and \(\frac{3}{(x+5)(x-6)}\) becomes \(\frac{3(x-5)}{(x-5)(x+5)(x-6)}\).
04

Perform the Addition and Subtraction

Now, we can perform the addition and subtraction: \(\frac{5(x-6)}{(x-5)(x+5)(x-6)} + \frac{4(x+5)}{(x-5)(x+5)(x-6)} - \frac{3(x-5)}{(x-5)(x+5)(x-6)}\). Since each expression has the same LCD, simply combine like terms from each of the numerators.
05

Simplify the Fraction

Finally, combine like terms and simplify your fraction: \(\frac{5x - 30 + 4x + 20 - 3x + 15}{(x-5)(x+5)(x-6)}\). This simplifies to \(\frac{6x + 5}{(x-5)(x+5)(x-6)}\).

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Most popular questions from this chapter

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\text { Simplify: } \frac{(y+2) y-2 \cdot 4}{4 y(y+4)}$$

What are similar triangles?

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{b}{a c+a d-b c-b d}-\frac{a}{a c+a d-b c-b d}$$

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)
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