[tex] \frac{x + 1}{x} = 4[/tex]
[tex]x + 1 = 4x[/tex]
[tex]x - 4x = - 1[/tex]
[tex] - 3x = - 1 [/tex]
[tex]x = \frac{ - 1}{ - 3} = \frac{1}{3} [/tex]
[tex]( \frac{x + 1}{x} )^{2} = {4}^{2} [/tex]
[tex] \frac{ {x}^{2} + 2x + 1 }{ {x}^{2} } = 16 [/tex]
[tex] \frac{ {x}^{2} + 1}{ {x}^{2} } + \frac{2x}{ {x}^{2} } = 16[/tex]
[tex] \frac{2x}{ {x}^{2} } = \frac{2 \times \frac{1}{3} }{( \frac{1}{3})^{2} } = \frac{2}{ \frac{1}{3} } = 6[/tex]
[tex] \frac{ {x}^{2} + 1 }{ {x}^{2} } + 6 = 16[/tex]
[tex] \frac{ {x}^{2} + 1 }{ {x}^{2} } = 10[/tex]
[tex]( \frac{x^{2} + 1 }{x ^{2} } )^{2} = {10}^{2} [/tex]
[tex] \frac{ {x}^{4} + 2 {x}^{2} + 1}{ {x}^{4} } = 100[/tex]
[tex] \frac{ {x}^{4} + 1 }{ {x}^{4} } + \frac{2x^{2} }{ {x}^{4} } = 100[/tex]
[tex] \frac{2 {x}^{2} }{ {x}^{4} } = \frac{2 \times {( \frac{1}{3} )}^{2} }{ {( \frac{1}{3} )}^{4} } = \frac{2}{ {( \frac{1}{3}) }^{2} } = \frac{2}{ \frac{1}{9} } = 18[/tex]
[tex] \frac{ {x}^{4} + 1 }{ {x}^{4} } + 18 =100 [/tex]
[tex] \frac{ {x}^{4} + 1 }{ {x}^{4} } = 82[/tex]
