SOLUTION: Tutor, I was wondering if I solved these problems correctly would you be able to help me? 1.) Solve: x/(5x+10) + (x-3)/(x+2) = 7/5 is the answer -17 ? 2.) write in

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Tutor, I was wondering if I solved these problems correctly would you be able to help me? 1.) Solve: x/(5x+10) + (x-3)/(x+2) = 7/5 is the answer -17 ? 2.) write in       Log On


   

Question 99549: Tutor,
I was wondering if I solved these problems correctly would you be able to help me?
1.) Solve: x/(5x+10) + (x-3)/(x+2) = 7/5 is the answer -17 ?
2.) write in simplest form: x^2 - 12x + 27 / x^2 - 81 is the answer x+3/x-9?
I am not sure I solved these correctly so any help is appreciated. Thank you.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Both problems need a little work.
.
First problem:
.
Given:
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x%2F%285x%2B10%29+%2B+%28x-3%29%2F%28x%2B2%29+=+7%2F5
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If you factor the denominator of the first term you get:
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x%2F%285%28x%2B2%29%29+%2B+%28x-3%29%2F%28x%2B2%29+=+7%2F5
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Next, give the second term the same denominator as the first term by multiplying it by 5.
You must also multiply the numerator of this second term by 5. The result is that the
equation then becomes:
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x%2F%285%28x%2B2%29%29+%2B+%285%2A%28x-3%29%29%2F%285%2A%28x%2B2%29%29+=+7%2F5
.
Now, both terms on the left side have the same denominator, so you can add their numerators
and place that sum over the common denominator to get:
.
%28x+%2B+5%2A%28x-3%29%29%2F%285%2A%28x%2B2%29%29+=+7%2F5
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Do the distributed multiplication in the numerator by multiplying 5 times each of the two
terms in the parentheses to get:
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%28x+%2B+5x+-+15%29%2F%285%2A%28x%2B2%29%29+=+7%2F5
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In the numerator, combine the terms containing x to get:
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%286x+-+15%29%2F%285%2A%28x%2B2%29%29+=+7%2F5
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Get rid of the denominator on the left side by multiplying both sides of this equation by
5%2A%28x%2B2%29. When you do that the equation becomes:
.
6x+-+15+=+%287%2F5%29%2A%285%29%2A%28x%2B2%29
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Notice on the right side that the multiplier of 5 is in the numerator and it will cancel
with the 5 in the denominator of the 7%2F5. As a result the equation reduces to:
.
6x+-+15+=+7%2A%28x%2B2%29
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Do the distributed multiplication on the right side by multiplying 7 times each of the
terms in the parentheses to get:
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6x+-+15+=+7x+%2B+14
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Get rid of the 7x on the right side by subtracting 7x from both sides. This changes the
equation to:
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-x+-+15+=+14
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Get rid of the -15 on the left side by adding +15 to both sides. This converts the equation
to:
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-x+=+29
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Finally, solve for +x by either dividing both sides by -1 or multiplying both sides by
-1 to get the answer: x+=+-29
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In the second problem, I suspect that you might have made a mistake in factoring the numerator
because you were very close to the right answer. You were given:
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%28x%5E2+-+12x+%2B+27%29+%2F+%28x%5E2+-+81%29
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Both the numerator and the denominator will factor, and you can then cancel the factors that
are common to the numerator and the denominator. When you do the factoring you get:
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%28%28x-9%29%2A%28x-3%29%29%2F%28%28x%2B9%29%2A%28x-9%29%29
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[You can multiply (x-9) times (x-3) to convince yourself that the product of these two
factors is x%5E2+-+12x+%2B+27]
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Now cancel the common factors:
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%28%28cross%28x-9%29%29%2A%28x-3%29%29%2F%28%28x%2B9%29%28cross%28x-9%29%29%29
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and you are left with the answer of:
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%28x-3%29%2F%28x%2B9%29
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This differs from your answer only by the minus sign in the numerator and the plus sign
in the denominator.
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Hope this helps you to identify your minor errors that you made in these two problems.
You seem to have the right idea of what to do but are just making minor mistakes which you
will be able to eliminate with a little more practice. Keep up the good work.
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