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`

`=>` 2y² + 2 = 5y

`=>` 2y² – 5y + 2 = 0

`=>` 2y² – 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 y = `1/2` 

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

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

`=>` x = 1 or 5x = –1

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