SOLUTION: 1) x^2/(x-3)=(x+2)/(2x-5) 2) (1/x)<[1/(x-3)] Please help me... Thank you!

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Question 1187209: 1) x^2/(x-3)=(x+2)/(2x-5)
2) (1/x)<[1/(x-3)]
Please help me... Thank you!
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
1) x^2/(x-3)=(x+2)/(2x-5)
x^2*(2x-5) = (x-3)*(x+2)
2x^3 - 5x^2 = x^2 - x - 6
2x^3 - 6x^2 + x + 6 = 0
2 is a zero, divide by (x-2)
(2x^3 - 6x^2 + x + 6)/(x-2) = 2x^2 - 2x - 3 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-2x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A2%2A-3=28.

Discriminant d=28 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+28+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+28+%29%29%2F2%5C2+=+1.8228756555323
x%5B2%5D+=+%28-%28-2%29-sqrt%28+28+%29%29%2F2%5C2+=+-0.822875655532295

Quadratic expression 2x%5E2%2B-2x%2B-3 can be factored:
2x%5E2%2B-2x%2B-3+=+%28x-1.8228756555323%29%2A%28x--0.822875655532295%29
Again, the answer is: 1.8228756555323, -0.822875655532295. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-2%2Ax%2B-3+%29



2) (1/x)<[1/(x-3)]
multiply by x(x-3)
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x-3 < x
-3 < 0
x <> 0 and x <> 3 ---> zero in a denominator
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x is any other number