Question

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

Answer 1

x = 5 or x = - $\frac{1}{5}$

simplify the left side by combining the fractions

$\frac{x + 1}{x - 1}$ - $\frac{x - 1}{x + 1}$

= $\frac{(x + 1)^{2} - (x - 1)^{2} }{(x - 1)(x + 1)}$

= $\frac{x^{2} + 2x + 1 - x^{2} + 2x - 1 }{(x - 1)(x + 1)}$ = $\frac{4x}{(x - 1)(x + 1)}$

$\frac{4x}{(x - 1)(x + 1)}$ = $\frac{5}{6}$ ( 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 = - $\frac{1}{5}$