Solve: `2((2x-1)/(x+3))-3((x+3)/(2x-1))=5,xne-3,(1)/(2).`

Explanation

Given equation is
`2((2x-1)/(x+3))-3((x+3)/(2x-1))=5" ".....(i)`
Let `(2x-1)/(x+3)=y.....(ii)`
Hence, `(x+3)/(2x-1)=(1)/(y)`
Now from equation (1)
`2y-(3)/(y)=5`
`implies2y^(2)-3=5y`
`implies2y^(2)-5y-3=0`
`implies2y^(2)-(6-1)y-3=0`
`implies2y^(2)-6y+1y-3=0`
`implies2y(y-3)+1(y-3)=0`
`implies(2y+1)(y-3)=0`
`implies2y+1=0ory-3=0`
when `implies2y+1=0impliesy=-(1)/(2)`
and `y-3=0impliesy=3`
Substituting values of y in equation (2)
when `y=-(1)/(2)`
`(2x-1)/(x+3)=-(1)/(2)`
`implies2(2x-1)=-1(x+3)implies4x-2=-x-3`
`implies5x=-1impliesx=-(1)/(5)`
when y=3
`(2x-1)/(x+3)=3`
`implies2x-1=3(x+3)implies2x-1=3x+9`
`implies-x=10impliesx=-10`
Hence, `x=-10orx=-(1)/(5)` are roots of the equation.