Question

Solve `((3x + 1)/(x + 1)) + ((x + 1)/(3x + 1)) = 5/2`

Answer 1

`((3x + 1)/(x + 1)) + ((x + 1)/(3x + 1)) = 5/2`

Let `(3x + 1)/(x + 1) = y`

then `y + 1/y = 5/2`

`=> (y^2 + 1)/y = 5/2`

`=> 2y^2 + 2 = 5y`

`=> 2y^2 - 5y + 2 = 0`

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

=> 2y(y - 2) - 1(y - 2) = 0

=> (y - 2)(2y - 1)  = 0

if y - 2 = 0 or 2y - 1 = 0

then y = 2 or 2y - 1 = 0

`=> (3x + 1)/(x + 1) = 2   or (3x + 1)/(x + 1) = 1/2`

`=> 3x + 1 = 2x + 2  or 6x + 2 = x + 1`

`=> x = 1 or x = (-1)/5`