Question

Solve (3/x-3) =(x/x-3)-(3/2) for x and determine if the solution is extraneous or not

Answer 1

3/(x-3)=x/(x-3)-3/2
6/2(x-3)=2x/2(x-3)-3(x-3)/2(x-3)
6=2x-3(x-3)
6=2x-3x+9
x=3

x=3 makes the denominator x-3 a zero, so the solution is extraneous,

Answer 2

given solution x =3 is extraneous.

Explanation:

`(3)/(x-3) =(x)/(x-3) -(3)/(2) \\ (3)/(x-3) =[tex](3)/(x-3) =(2x-3(x-3))/(2(x-3))\\=(2x-3x+9))/(2(x-3))\\= (9-x)/(2(x-3)) \\\\(3)/(x-3) = (9-x)/(2(x-3)) \\\\  \\ 6(x-3) = (9-x)(x-3) \\6(x-3) - (9-x)(x-3) = 0 \\(x-3) (6-9+x) = 0 \\(x-3) (x-3) = 0x=3` [taking LCM right hand side] ,we get

gives x =3 as a solution which makes the original equation undefined

therefore the solution extraneous .