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Solve `((2x - 3)/(X -1)) - 4((X - 1)/(2x - 3)) = 3` - Mathematics

Solve `((2x - 3)/(x -1)) - 4((x - 1)/(2x - 3)) = 3`

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Solution

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

Let `(2x - 3)/(x - 1) = y`

`=> y - 4/y = 3`

`=> y^2 - 4 = 3y`

`=> y^2 - 3y - 4 = 0`

`=> y^2 - 4y + y - 4 = 0`

`=> (y - 4)(y + 1) = 0`

if y - 4 = 0 or y + 1 = 0

then y = 4 or y + 1 = 0

`=> (2x - 3)/(x - 1) = 4` or `(2x - 3)/(x- 1) = -1`

`=> 4x - 4 = 2x - 3 or 2x - 3 = -x + 1`

=> 2x = 1 or 3x = 4

`=> x = 1/2  or x = 4/3 = 1 1/3`

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