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Solve the equation. Check all proposed solutions. Show work in solving and in checking.

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Solve the equation. Check all proposed solutions. Show work in solving and in checking.

  x         x       102
----  +  ---- =  -----       
x-3      x+3     x2-9
 
First let's add the fractions on the left.  To add any fraction, you must put them over a common denominator, in this case (x-3)(x+3)
 
   x   (x+3)          x   (x-3)
------ ------  +  ------ -------
(x-3) (x+3)      (x+3) (x-3)
 
Multiply out the numerator and denominator then combine like terms
 
x2 + 3x + x2 - 3x          2x2
---------------------  =   ------  
       x2 - 9                  x2-9
 
Now we can re-state the original problem as:
 
2x2/(x2-9) = 102/(x2-9)
 
2x2 = 102      (Multiply both sides by (x2-9) )
 
x2 = 51         (Divide both sides by 2)
 
x = ±√51      (Take the square root of both sides)
 
To check, plug x = +√51 and x = -√51 into your original equation (below) and make sure the left hand side equals the right hand side.  If it does, it's a solution.
 
 2x2/(x2-9) = 102/(x2-9)

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