SOLUTION: solve: 9/x-5 -1=8/x+5

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Question 94432: solve:
9/x-5 -1=8/x+5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
9%2F%28x-5%29+-1=8%2F%28x%2B5%29


%28x-5%29%28x%2B5%29%289%2F%28x-5%29+-1%29=%28x-5%29%28x%2B5%29%288%2F%28x%2B5%29%29 Multiply by the LCD %28x-5%29%28x%2B5%29

9%28x%2B5%29+-%28x-5%29%28x%2B5%29=8%28x-5%29 Distribute and simplify

9x%2B45+-%28x-5%29%28x%2B5%29=8x-40 Distribute again

9x%2B45+-%28x%5E2-25%29=8x-40 Foil

9x%2B45+-x%5E2%2B25=8x-40 Distribute the negative

9x%2B45+-x%5E2%2B25-8x%2B40=0 Get all terms to one side

-x%5E2%2Bx%2B110=0 Combine like terms



Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve -x%5E2%2Bx%2B110=0 ( notice a=-1, b=1, and c=110)

x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A-1%2A110+%29%29%2F%282%2A-1%29 Plug in a=-1, b=1, and c=110



x+=+%28-1+%2B-+sqrt%28+1-4%2A-1%2A110+%29%29%2F%282%2A-1%29 Square 1 to get 1



x+=+%28-1+%2B-+sqrt%28+1%2B440+%29%29%2F%282%2A-1%29 Multiply -4%2A110%2A-1 to get 440



x+=+%28-1+%2B-+sqrt%28+441+%29%29%2F%282%2A-1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-1+%2B-+21%29%2F%282%2A-1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-1+%2B-+21%29%2F-2 Multiply 2 and -1 to get -2

So now the expression breaks down into two parts

x+=+%28-1+%2B+21%29%2F-2 or x+=+%28-1+-+21%29%2F-2

Lets look at the first part:

x=%28-1+%2B+21%29%2F-2

x=20%2F-2 Add the terms in the numerator
x=-10 Divide

So one answer is
x=-10



Now lets look at the second part:

x=%28-1+-+21%29%2F-2

x=-22%2F-2 Subtract the terms in the numerator
x=11 Divide

So another answer is
x=11

So our solutions are:
x=-10 or x=11

Notice when we graph -x%5E2%2Bx%2B110, we get:

+graph%28+500%2C+500%2C+-20%2C+21%2C+-20%2C+21%2C-1%2Ax%5E2%2B1%2Ax%2B110%29+

and we can see that the roots are x=-10 and x=11. This verifies our answer



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Check:
Let's check the solution x=-10

9%2F%28x-5%29+-1=8%2F%28x%2B5%29 Start with the given equation

9%2F%28-10-5%29+-1=8%2F%28-10%2B5%29 Plug in x=-10

9%2F%28-15%29+-1=8%2F%28-5%29 Simplify the denominators

-3%2F5+-1=-8%2F5 Reduce


-3%2F5+-5%2F5=-8%2F5 Rewrite -1 as -5%2F5

-8%2F5=-8%2F5 Combine the fractions. Since the two sides of the equation are equal, this verifies the solution x=-10







Let's check the solution x=11

9%2F%28x-5%29+-1=8%2F%28x%2B5%29 Start with the given equation

9%2F%2811-5%29+-1=8%2F%2811%2B5%29 Plug in x=11

9%2F%286%29+-1=8%2F%2816%29 Simplify the denominators

3%2F2+-1=1%2F2 Reduce


3%2F2+-2%2F2=1%2F2 Rewrite -1 as -2%2F2

1%2F2=1%2F2 Combine the fractions. Since the two sides of the equation are equal, this verifies the solution x=11