Question

Solve the following equation by using formula :

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

Answer 1

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

`(2x + 2 - x - 2)/((x + 2)(x + 1)) = (4x + 12 - 3x - 12)/((x + 4)(x + 3)`

⇒ `x/((x + 2)(x + 1)) = x/((x + 4)(x + 3)`

⇒ `(1)/((x + 2)(x + 1)) = (1)/((x + 4)(x + 3)` ...[Dividing by x if x ≠ 0]

⇒ `(1)/(x^2 + 3x + 2) = (1)/(x^2 + 7x + 12)`

⇒ x² + 7x + 12 – x2 – 3x – 2 = 0
⇒ 4x + 10
⇒ 2x + 5 = 0
⇒ 2x = –5

⇒ x = `(-5)/(2)`
If x = 0, then

`(0)/((x + 2)(x + 1)) = (0)/((x + 4)(x + 3)`
Which is correct

Hence x = 0, `(-5)/(2)`.