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Chapter 7: Problem 90

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{7 y-2}{y^{2}-y-12}+\frac{2 y}{4-y}+\frac{y+1}{y+3}$$

Short Answer

Expert verified
The simplified result of the operation is \(\frac{9 y - 6}{(y - 4)(y + 3)}\)

Step by step solution

01

Factor the Quadratic Polynomial

Factor the quadratic polynomial \(y^{2}-y-12\) in the denominator of the first fraction to obtain \((y-4)(y+3)\).
02

Rewrite the Denominators

Rewrite the denominator of the second fraction to match the standard form \(y-n\). So, \(4-y\) becomes \(-(y-4)\). Rewrite the denominator of the third fraction as \((y+3)\).
03

Find the Common Denominator

Identify the common denominator for the three fractions, which would be \(-(y-4)(y+3)\).
04

Rewrite the Numerators

Rewrite the numerators by multiplying by the necessary factors such that all fractions have the common denominator. For the second fraction, a negative is introduced due to the manipulation in step 2.
05

Perform the Addition

Perform the addition by combining the numerators over the common denominator.
06

Simplify the Result

Simplify the resultant fraction and eliminate any common factor in the numerator and denominator if available.

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x}+\frac{4}{x-6}$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8}-\frac{5}{6}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x}+9$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$

Factor completely: \(81 x^{4}-1\)
See all solutions

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