Answer:
[tex]\boxed{x=5},x=-9[/tex]
Step-by-step explanation:
To rid the denominators, multiply both sides by [tex](x-3)(x+1)[/tex].
We get:
[tex]6(x+1)-12(x-3)=(x-3)(x+1)[/tex]
Simplifying, we have:
[tex]6x+6-12x+36=x^2-3x+x-3,\\-6x+42=x^2-2x-3,\\x^2+4x-45=0[/tex]
Solving, we get:
[tex]x^2+4x-45=0,\\(x-5)(x+9)=0,\\x=5,x=-9[/tex]
Therefore, the solution with the greatest value is [tex]x=\boxed{5}[/tex]
