Question

Solve:

`2((2x - 1)/(x + 3)) - 3((x + 3)/(2x - 1)) = 5; x ≠ -3, (1)/(2)`

Answer 1

Given `2((2x - 1)/(x + 3)) - 3((x + 3)/(2x - 1)) = 5`       ...(1)

Put `(2x - 1)/(x + 3) = y` in equation (1); we have

`2y - (3)/y = 5`

⇒ 2y² – 5y – 3 = 0

⇒ 2y² – 6y + y – 3 = 0

⇒ 2y(y – 3) + 1(y – 3) = 0

⇒ (y – 3)(2y + 1) = b

Either y – 3 = 0 or 2y + 1 = 0

⇒ y = 3 or y = `-1/2` 

⇒ `(2x - 1)/(x + 3) = 3` or `(2x - 1)/(x + 3) = -1/2`

⇒ 2x – 1 = 3x + 9

or 2(2x – 1) = – (x + 3)

⇒ 3x – 2x = – 1 – 9

or 4x + x = 2 – 3

⇒ x = –10 or 5x = –1

i.e. `x = -1/5`

Thus, `x = -10, -1/5`