SOLUTION: Hello, I'm having a bit of a difficult time figuring out if I have done this problem correctly, and would greatly appreciate if someone could tell me if I've solved it correctl

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: Hello, I'm having a bit of a difficult time figuring out if I have done this problem correctly, and would greatly appreciate if someone could tell me if I've solved it correctl      Log On


   

Question 88323: Hello,
I'm having a bit of a difficult time figuring out if I have done this problem correctly, and would greatly appreciate if someone could tell me if I've solved it correctly or if I am way off track, and show me what I need to do.
Here is the original problem that I have to solve: 9/(x-5) - 1 = 8/(x+5).
Here's the solution that I've come to: First, I moved 8/x+5 to the left, so that the equation read like this: 9/x-5 -1 -8/x+5 = 0. Then, I moved -1 to the right of the equation, so that it read like this: 9/x-5 - 8/x+5 -1 =0, which I think is the solution to the problem, right? Is there further solving or simplifying involved that I'm overlooking or is my solution correct?
Thank you very, very much in advance for your help!

Found 2 solutions by stanbon, Earlsdon:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
In general, when you have an equation with denominators, get rid of them.
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Multiply thru by (x-5)(x+5) to get:
9/(x-5) - 1 = 8/(x+5)
9(x+5) - (x-5)(x+5) = 8(x-5)
9x+45 - (...)(...) = 8x-40
x + 85 -[x^2-25] = 0
-x^2+x+110 = 0
x^2 -x -110 = 0
(x-11)(x+10) = 0
x = 11 or x = -10
===========================
Cheers,
Stan H.

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Ok, your first move was good! So you have:
9%2F%28x-5%29+-+8%2F%28x%2B5%29+-+1+=+0 Now add 1 (move the -1 to the right side) to both sides.
9%2F%28x-5%29+-+8%2F%28x%2B5%29+=+1 Now add the two fractions on the left side. The LCD is %28x-5%29%28x%2B5%29 so you now have:
%289%28x%2B5%29-8%28x-5%29%29%2F%28x-5%29%28x%2B5%29+=+1 Simplify the numerator.
%289x%2B45-8x%2B40%29%2F%28x-5%29%28x%2B5%29+=+1 Continue to simplify.
%28x%2B85%29%2F%28x-5%29%28x%2B5%29+=+1 Now multiply both sides by %28x-5%29%28x%2B5%29 to get:
x%2B85+=+%28x-5%29%28x%2B5%29 Expand the right side.
x%2B85+=+x%5E2-25 Now subtract x%2B85 from both sides.
0+=+x%5E2-x-110 or x%5E2-x-110+=+0 This can be factored:
%28x%2B10%29%28x-11%29+=+0 Apply the zero products principle to get:
x+=+-10 or x+=+11 These are the solutions!
Let's check the solutions by substituting the solutions, one-by-one into the original equation:
%289%2F%28x-5%29%29-1+=+8%2F%28x%2B5%29 Substitute x = -10.
%289%2F%28-10-5%29%29-1+=+8%2F%28-10%2B5%29
%289%2F%28-15%29%29+-+1+=+8%2F%28-5%29
%289%2F%28-15%29%29+-+%28%28-15%29%2F%28-15%29%29+=+8%2F%28-5%29
-24%2F15+=+-%288%2F5%29 Reduce the fraction on the left side.
-%288%2F5%29+=+-%288%2F5%29 It checks!
Try the other solution yourself and see how you do.