What is x/((x+1))> 1/x ?

The solution to x/((x+1))> 1/x is x (1+sqrt(5))/2

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x/((x+1))> 1/x

\frac{x}{( x + 1 )} > \frac{1}{x}

x < - 1 or \frac{1 - 5}{2} < x < 0 or x > \frac{1 + 5}{2}

Interval Notation

(- \infty, - 1) \cup (\frac{1 - 5}{2}, 0) \cup (\frac{1 + 5}{2}, \infty)

Decimal Notation

x < - 1 or - 0.61803 \dots < x < 0 or x > 1.61803 \dots

Solution steps

\frac{x}{x + 1} > \frac{1}{x}

Rewrite in standard form

\frac{x ^{2} - x - 1}{x ( x + 1 )} > 0

Factor

x^{2} - x - 1 : (x - \frac{1 + 5}{2}) (x - \frac{1 - 5}{2})

\frac{( x - \frac{1 + 5}{2} ) ( x - \frac{1 - 5}{2} )}{x ( x + 1 )} > 0

Identify the intervals

x < - 1 or \frac{1 - 5}{2} < x < 0 or x > \frac{1 + 5}{2}

\frac{6 x ( 12 + 8 )}{5} > 2 x^{2}

\frac{( ( x + 3 ) ^{2} \cdot ( x ^{2} + x + 1 ) )}{( ( 4 - x ) \cdot x )} \geq 0

\frac{( x ^{2} - 5 x + 6 )}{( x - 2 )} < 0

x^{2} + 6 x + 15 > 0

(x + 1 + 5)^{2} + 5 \geq 3 - (x + 1 + 3)^{2}

What is x/((x+1))> 1/x ?
The solution to x/((x+1))> 1/x is x<-1\lor (1-sqrt(5))/2 <x<0\lor x>(1+sqrt(5))/2