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4/(x-2)=1+6/(x+2) ----(1)
\n" ); document.write( "Multiplying by (x-2)(x+2) (which is the lcm of (x-2) and (x+2) )
\n" ); document.write( "4(x+2)=(x-2)(x+2)+6(x-2)
\n" ); document.write( "4x+8 = x^2-2^2 +6x-12
\n" ); document.write( "4x+8 = x^2-4+6x-12
\n" ); document.write( "0 = x^2+(6x-4x)-4-12-8 (grouping like term, changing sign while changing side)
\n" ); document.write( "0 = x^2+2x-24
\n" ); document.write( "That is x^2+2x-24 = 0 ----(2)
\n" ); document.write( "x^2+(6x-4x)-24 = 0 (splitting the mid term into two parts so that their sum
\n" ); document.write( "is the mid term and their product is the product of the square term and the constant term)
\n" ); document.write( "(x^2+6x)-4x-24 = 0 (by additive associativity)
\n" ); document.write( "x(x+6)-4(x+6) =0
\n" ); document.write( "xp-4p = 0 where p = (x+6)
\n" ); document.write( "p(x-4) = 0
\n" ); document.write( "That is (x+6)(x-4) = 0 (putting p back)
\n" ); document.write( "(x+6) = 0 gives x =-6
\n" ); document.write( "(x-4) = 0 gives x = 4
\n" ); document.write( "Answer: x = -6 and x = 4
\n" ); document.write( "Verification: x = -6 in (1)
\n" ); document.write( "LHS = 4/(x-2)= 4/(-6-2) = 4/(-8) = (-1/2)
\n" ); document.write( "RHS = 1+6/(x+2) = 1+6/(-6+2)= 1+[6/(-4)] =1-3/2 = (-1/2) = LHS
\n" ); document.write( "x = 4 in (1)
\n" ); document.write( "LHS = 4/(x-2)= 4/(4-2) = 4/2 = 2
\n" ); document.write( "RHS = 1+6/(x+2) = 1+6/(4+2)= 1+6/6 =1+1 = 2 = LHS
\n" ); document.write( "Therefore our values are correct\r
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