Question

Solve for x : (x +1) / (x- 1) - (x - 1)/ (x + 1) = 5/6

Answer 1

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

simplify the left side by combining the fractions

`(x + 1)/(x - 1)` -
`(x - 1)/(x + 1)`

=
`((x + 1)^(2) - (x - 1)^(2) )/((x - 1)(x + 1))`

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

`(4x)/((x - 1)(x + 1))` =
`(5)/(6)` ( cross- multiply )

24x = 5 (x² - 1)

24x = 5x² - 5 ( rearrange into standard form )

5x² - 24x - 5 = 0 ← in standard form

consider the factors of the product 5 × - 5 = - 25 which sum to - 24

The factors are - 25 and + 1 ( split the middle term using these factors )

5x² - 25x + x - 5 = 0 ( factor by grouping )

5x( x - 5 ) + 1( x - 5 ) = 0 ( take out common factor (x - 5 ) )

(x - 5 )( 5x + 1 ) = 0

equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

5x + 1 = 0 ⇒ x = -
`(1)/(5)`