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

Explanation

On putting `(2x-1)/(x+3)=y`, the given equation becomes
`2y-(3)/(y)=5implies2y^(2)-3=5y`
`implies" "2y^(2)-5y-3=0implies2y^(2)-6y+y-3=0`
`implies" "2y(y-3)+(y-3)=0implies(y-3)(2y+1)=0`
`implies" "y-3=0" or "2y+1=0`
`impliesy=3" or "y=(-1)/(2).`
Case I `y=3implies(2x-1)/(x+3)=0`
`implies" "2x-1=3(x+3)" "["by cross multiplication"]`
`implies2x-1=3x+9`
`impliesx=-10.`
Case II `y=(-1)/(2)implies(2x-1)/(x+3)=(-1)/(2)`
`implies" "2(2x-1)=-(x+3)`
`implies" "5x=-1`
`impliesx=(-1)/(5).`
Hence, -10 and `(-1)/(5)` are the roots of the given equation.