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Question 30421: 4/x-2=1+6/x+2
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! 4/(x-2)=1+6/(x+2) ----(1)
Multiplying by (x-2)(x+2) (which is the lcm of (x-2) and (x+2) )
4(x+2)=(x-2)(x+2)+6(x-2)
4x+8 = x^2-2^2 +6x-12
4x+8 = x^2-4+6x-12
0 = x^2+(6x-4x)-4-12-8 (grouping like term, changing sign while changing side)
0 = x^2+2x-24
That is x^2+2x-24 = 0 ----(2)
x^2+(6x-4x)-24 = 0 (splitting the mid term into two parts so that their sum
is the mid term and their product is the product of the square term and the constant term)
(x^2+6x)-4x-24 = 0 (by additive associativity)
x(x+6)-4(x+6) =0
xp-4p = 0 where p = (x+6)
p(x-4) = 0
That is (x+6)(x-4) = 0 (putting p back)
(x+6) = 0 gives x =-6
(x-4) = 0 gives x = 4
Answer: x = -6 and x = 4
Verification: x = -6 in (1)
LHS = 4/(x-2)= 4/(-6-2) = 4/(-8) = (-1/2)
RHS = 1+6/(x+2) = 1+6/(-6+2)= 1+[6/(-4)] =1-3/2 = (-1/2) = LHS
x = 4 in (1)
LHS = 4/(x-2)= 4/(4-2) = 4/2 = 2
RHS = 1+6/(x+2) = 1+6/(4+2)= 1+6/6 =1+1 = 2 = LHS
Therefore our values are correct
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