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Chapter 7: Problem 89

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{9 y+3}{y^{2}-y-6}+\frac{y}{3-y}+\frac{y-1}{y+2}$$

Short Answer

Expert verified
The result of the operation is \( \frac{-2y^2+10y+6}{(y-3)(y+2)} \)

Step by step solution

01

Factorize the Denominators

Factorize the denominators of the fractions to find a common denominator:\[\frac{9 y+3}{(y-3)(y+2)}+\frac{y}{3-y}+\frac{y-1}{y+2}\]Note: \(3-y\) is equal to \(-(y-3)\). Therefore, the second fraction can be written as:\[\frac{9 y+3}{(y-3)(y+2)}-\frac{y}{y-3}+\frac{y-1}{y+2}\]
02

Find a Common Denominator and Rewrite the Fractions

Find a common denominator, which is \((y-3)(y+2)\). Rewrite the fractions so they all have this common denominator:\[\frac{9 y+3}{(y-3)(y+2)}-\frac{(y)(y+2)}{(y-3)(y+2)}+\frac{(y-1)(y-3)}{(y-3)(y+2)}\]
03

Combine Like Terms

Add and subtract the numerators over the common denominator:\[\frac{9y+3-y^2-2y-y^2+3y+3}{(y-3)(y+2)}\]Which simplifies to:\[\frac{-2y^2+10y+6}{(y-3)(y+2)}\]
04

Simplify the Result

The result is a simple fraction which is the solution:\[\frac{-2y^2+10y+6}{(y-3)(y+2)}\]

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\begin{aligned} &\frac{x^{2}+9 x}{4 x^{2}-11 x-3}+\frac{3 x-5 x^{2}}{4 x^{2}-11 x-3}\\\ &x^{2}-4 x-4 x-4 \end{aligned}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y^{2}-2}{3 y^{2}+10 y-8}-\frac{y+10}{3 y^{2}+10 y-8}-\frac{y^{2}-6 y}{3 y^{2}+10 y-8}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+3}{x^{2}-x-30}+\frac{x-2}{30+x-x^{2}}$$

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