x = 5 or x = - [tex]\frac{1}{5}[/tex]
simplify the left side by combining the fractions
[tex]\frac{x + 1}{x - 1}[/tex] - [tex]\frac{x - 1}{x + 1}[/tex]
= [tex]\frac{(x + 1)^{2} - (x - 1)^{2} }{(x - 1)(x + 1)}[/tex]
= [tex]\frac{x^{2} + 2x + 1 - x^{2} + 2x - 1 }{(x - 1)(x + 1)}[/tex] = [tex]\frac{4x}{(x - 1)(x + 1)}[/tex]
[tex]\frac{4x}{(x - 1)(x + 1)}[/tex] = [tex]\frac{5}{6}[/tex] ( 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 = - [tex]\frac{1}{5}[/tex]
