Question

ctions: simplify each expression. find you answer on the back and color the mbered spaces using the indicated color.

1. $\frac{n+2}{4n-4} + \frac{6}{4n-4}$
2. $\frac{2n}{n^2-4} - \frac{n+6}{(n+2)(n-2)}$
3. $\frac{n+1}{(n-5)(n+3)} - \frac{n-3}{n^2-2n-15}$
4. $5 + \frac{5n}{4n-24}$
5. $\frac{n-5}{2n+8} - \frac{5}{2n}$
6. $\frac{n-4}{n-6} + \frac{4n}{3}$
7. $\frac{4}{n-5} - \frac{2n}{n+5}$
8. $\frac{6n}{n+2} - \frac{6}{n-1}$
9. $\frac{4}{3n-3} + \frac{6n}{3n+6}$

Explanation:

Step1: Combine like denominators

$\frac{n+2}{4n-4} + \frac{6}{4n-4} = \frac{(n+2)+6}{4n-4}$

Step2: Simplify numerator and denominator

$\frac{n+8}{4(n-1)}$

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Step1: Factor denominators

$\frac{2n}{(n-2)(n+2)} - \frac{n+6}{(n+2)(n-2)}$

Step2: Combine fractions

$\frac{2n-(n+6)}{(n-2)(n+2)}$

Step3: Simplify numerator

$\frac{n-6}{(n-2)(n+2)}$

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Step1: Factor second denominator

$\frac{n+1}{(n-5)(n+3)} - \frac{n-3}{(n-5)(n+3)}$

Step2: Combine fractions

$\frac{(n+1)-(n-3)}{(n-5)(n+3)}$

Step3: Simplify numerator

$\frac{4}{(n-5)(n+3)}$

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Step1: Factor denominator

$5 + \frac{5n}{4(n-6)}$

Step2: Get common denominator

$\frac{5 \cdot 4(n-6)}{4(n-6)} + \frac{5n}{4(n-6)}$

Step3: Expand and simplify numerator

$\frac{20(n-6)+5n}{4(n-6)} = \frac{20n-120+5n}{4(n-6)} = \frac{25n-120}{4(n-6)}$

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Step1: Factor first denominator

$\frac{n-5}{2(n+4)} - \frac{5}{2n}$

Step2: Get common denominator

$\frac{(n-5)n}{2n(n+4)} - \frac{5(n+4)}{2n(n+4)}$

Step3: Expand and simplify numerator

$\frac{n^2-5n-5n-20}{2n(n+4)} = \frac{n^2-10n-20}{2n(n+4)}$

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Step1: Get common denominator

$\frac{3(n-4)}{3(n-6)} + \frac{4n(n-6)}{3(n-6)}$

Step2: Expand and simplify numerator

$\frac{3n-12+4n^2-24n}{3(n-6)} = \frac{4n^2-21n-12}{3(n-6)}$

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Step1: Get common denominator

$\frac{4(n+5)}{(n-5)(n+5)} - \frac{2n(n-5)}{(n-5)(n+5)}$

Step2: Expand and simplify numerator

$\frac{4n+20-2n^2+10n}{(n-5)(n+5)} = \frac{-2n^2+14n+20}{(n-5)(n+5)} = \frac{-2(n^2-7n-10)}{(n-5)(n+5)}$

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Step1: Get common denominator

$\frac{6n(n-1)}{(n+2)(n-1)} - \frac{6(n+2)}{(n+2)(n-1)}$

Step2: Expand and simplify numerator

$\frac{6n^2-6n-6n-12}{(n+2)(n-1)} = \frac{6n^2-12n-12}{(n+2)(n-1)} = \frac{6(n^2-2n-2)}{(n+2)(n-1)}$

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Step1: Factor denominators

$\frac{4}{3(n-1)} + \frac{6n}{3(n+2)}$

Step2: Get common denominator

$\frac{4(n+2)}{3(n-1)(n+2)} + \frac{6n(n-1)}{3(n-1)(n+2)}$

Step3: Expand and simplify numerator

$\frac{4n+8+6n^2-6n}{3(n-1)(n+2)} = \frac{6n^2-2n+8}{3(n-1)(n+2)}$

Answer:

1. $\boldsymbol{\frac{n+8}{4(n-1)}}$
2. $\boldsymbol{\frac{n-6}{(n-2)(n+2)}}$
3. $\boldsymbol{\frac{4}{(n-5)(n+3)}}$
4. $\boldsymbol{\frac{25n-120}{4(n-6)}}$
5. $\boldsymbol{\frac{n^2-10n-20}{2n(n+4)}}$
6. $\boldsymbol{\frac{4n^2-21n-12}{3(n-6)}}$
7. $\boldsymbol{\frac{-2n^2+14n+20}{(n-5)(n+5)}}$
8. $\boldsymbol{\frac{6n^2-12n-12}{(n+2)(n-1)}}$
9. $\boldsymbol{\frac{6n^2-2n+8}{3(n-1)(n+2)}}$