1. rational expressions

if someone could help me with these seven HW problems. i had do do 75 of them, i just cant figure out these 7! help is much needed..thanks you so much!

Simplify
#1. 6t^2-7t-5 all over 2t+1

#2 q^2-11q+30 Divided by q^2-2q-24
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2q^2-7q-15 q^2-q-20

#3 4x+12 over 3-x minus 3x+15 over 3-x

#4 x+3 over x^2-3x-10 minus 2 over x-5

#5 x^-2-y^-2 over y^-1-x^-1

Solve the Equation

#6 3x over x^2-9 Plus 1 over x-3 = 3 over x+3

Word Problem

#7 Following a severe snowstorm, Ken and Bettina must clear their driveway and sidewalk. Ken can clear the snow by himself in 4 hours, and bettina can clear the snow be herself in 6 hours. After bettina has been working for 3 hours, ken is able to join her. how much longer will it take them working together to remove the rest of the snow.

Any help would be much appreciated. Thank You!

2. Originally Posted by mathnoobie25
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Simplify
#1. 6t^2-7t-5 all over 2t+1

...
Hello,

$\displaystyle \frac{6t^2-7t-5}{2t+1}=\frac{(2t+1)(3t-5)}{2t+1}=3t-5\ , \ t \neq -\frac{1}{2}$

EB

3. Originally Posted by mathnoobie25
...
Simplify

#2 q^2-11q+30 Divided by q^2-2q-24
------------- -----------
2q^2-7q-15 q^2-q-20

...
Hello,

$\displaystyle \frac{q^2-11q+30}{2q^2-7q-15} \div \frac{q^2-2q-24}{q^2-q-20}=$

$\displaystyle \frac{(q-6)(q-5)}{(q-5)(2q+3)} \div \frac{(q-6)(q+4)}{(q-5)(q+4)}=$. Now transform into a product:

$\displaystyle \frac{(q-6)(q-5)}{(q-5)(2q+3)} \cdot\frac{(q-5)(q+4)}{(q-6)(q+4)}= \frac{q-5}{2q+3}$

EB

4. Originally Posted by mathnoobie25
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Simplify

#3 4x+12 over 3-x minus 3x+15 over 3-x

...
Hello,

$\displaystyle \frac{4x+12}{3-x} - \frac{3x+15}{3-x}=\frac{x-3}{3-x}=-1$

EB

5. Originally Posted by mathnoobie25
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Simplify

#4 x+3 over x^2-3x-10 minus 2 over x-5
...
Hello,

$\displaystyle \frac{x+3}{x^2-3x-10} - \frac{2}{x-5}=\frac{x+3}{(x-5)(x+2)} - \frac{2}{x-5}=$ $\displaystyle \frac{x+3}{(x-5)(x+2)} - \frac{2(x+2)}{(x-5)(x+2)}$

$\displaystyle =\frac{-x-1}{(x-5)(x+2)}$

EB

6. Originally Posted by mathnoobie25
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Simplify

#5 x^-2-y^-2 over y^-1-x^-1

...
Hello,

$\displaystyle \frac{x^{-2} - y^{-2}}{y^{-1} - x^{-1}}=\frac{(x^{-1}-y^{-1})(x^{-1}+y^{-1})}{y^{-1} - x^{-1}}=$ $\displaystyle =-(x^{-1}+y^{-1})$

EB

7. Originally Posted by mathnoobie25
...

Solve the Equation

#6 3x over x^2-9 Plus 1 over x-3 = 3 over x+3

...
Hello,

I assume that you know: $\displaystyle x^2-9=(x+3)(x-3)$

$\displaystyle \frac{3x}{x^2-9}+\frac{1}{x-3}=\frac{3}{x+3}$. Multiply by (x^2-9) (it's the common denominator) and cancel the equal factors out(?):

$\displaystyle 3x+x+3=3(x-3)$. Solve for x. I've got x = -12.

EB

PS: to #7. It's raining here, no snow in sight - and by the way it is very late and I'm afraid that I'll make some silly mistakes. So I leave #7 for someone else.

8. Originally Posted by mathnoobie25
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Word Problem

#7 Following a severe snowstorm, Ken and Bettina must clear their driveway and sidewalk. Ken can clear the snow by himself in 4 hours, and bettina can clear the snow be herself in 6 hours. After bettina has been working for 3 hours, ken is able to join her. how much longer will it take them working together to remove the rest of the snow. ...
Hello and good morning,

let S be the total amount of snow which has to be removed

then
Ken removes $\displaystyle \frac{S}{4}$ in 1 hour
Bettina removes $\displaystyle \frac{S}{6}$ in 1 hour

The complete work could be described by:

$\displaystyle \underbrace{\frac{S}{6} \cdot 3}_{\text{ Bettina working alone}} + \underbrace{\left(\frac{S}{4}+\frac{S}{6} \right) \cdot x}_{\text{B.+K. working together}} =S$. Divide by S:

$\displaystyle \frac{1}{2}+\frac{5}{12}x=1$. Solve for x. I've got x = 1.2 hours = 1 hour 12 minutes.

EB