Question

How do you solve `\frac{3}{2x-4}-\frac{2}{x^{2}-x-2}=\frac{9}{x+1}`?

Answer 1

`x=7/3`

Explanation:

First we factor the denominators, then find a common denominator so we can subtract, then cross multiply and solve the resulting polynomial equation.

`frac{3}{2x-4} - frac{2}{x^2 - x -2 } = frac{9}{x+1}`

`frac{3}{2(x-2)} - frac{2}{(x+1)(x-2)} = frac{9}{x+1}`

`frac{3(x+1) - 2(2)}{2(x+1)(x-2)} = frac{9}{x+1}`

`frac{3x-1}{2(x+1)(x-2)} = frac{9}{x+1}`

`(3x-1)(x+1) = 18(x+1)(x-2)`

`(x+1)( (3x - 1) - 18(x-2) ) = 0`

`(x+1)(-15 x + 35) = 0`

`x = -1` or `x = 35/15 = 7/3`

Answer 2

`x=7/3`

Explanation:

`3/(2x-4)-2/(x^2-x-2)=9/(x+1)`

Factorising

`2x-4=2(x-2)`

`x^2-x-2=x^2-2x+x-2`

`=x(x-2)+(x-2)`

`=(x+1)(x-2)`

`3/(2x-4)-2/(x^2-x-2)=3/(2(x-2))-2/((x+1)(x-2))`

`3/(2(x-2))-2/((x+1)(x-2))=9/(x+1)`

`3/(2(x-2))-9/(x+1)=2/((x+1)(x-2))`

Multiplying throughout by `(x+1)(x-2)`

`3/2(x+1)-9(x-2)=2`

`3/2x+3/2-9x+18=2`

`(3/2-9)x+(3/2+18-2)=0`

`-15/2x+35/2=0`

`-15x+35=0`

`15x=35`

`x=35/15`

`x=7/3`

Check:

lhs=
`3/(2x-4)-2/(x^2-x-2)=3/(2xx7/3-4)-2/((7/3)^2-7/3-2)`

`=3/(14/3-4)-2/(49/9-7/3-2)=9/(14-12)-18/(49-21-18)`

`=9/2-18/10=45/10-18/10=(45-18)/10=27/10`

`rhs=9/(x+1)=9/(7/3+1)=27/(7+3)=27/10`

`lhs=rhs`