What is ((x-1))/x-((x+1))/((x-1))<2 ?

The solution to ((x-1))/x-((x+1))/((x-1)) 1

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((x-1))/x-((x+1))/((x-1))<2

\frac{( x - 1 )}{x} - \frac{( x + 1 )}{( x - 1 )} < 2

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

Interval Notation

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

Decimal Notation

x < - 1 or 0 < x < 0.5 or x > 1

Solution steps

\frac{x - 1}{x} - \frac{x + 1}{x - 1} < 2

Rewrite in standard form

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

Factor

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

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

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

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

\frac{1}{( 2 x - 3 )} \leq \frac{1}{( 2 x + 5 )}

\frac{( x - 4 ) ^{2}}{x} > 0

\frac{( 2 x - 7 )}{( x - 5 )} \leq 3

\frac{( ( x - 7 - x ^{2} ) ( x ^{2} + 6 ) )}{( x ^{2} - x - 1 )} \leq 0

x (x^{2} - x) (x^{3} - x) (x^{4} - x) < 0

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