What is 1/((x-1))-1/((x-2))+1/((x-3))>0 ?

The solution to 1/((x-1))-1/((x-2))+1/((x-3))>0 is 1 3

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1/((x-1))-1/((x-2))+1/((x-3))>0

\frac{1}{( x - 1 )} - \frac{1}{( x - 2 )} + \frac{1}{( x - 3 )} > 0

1 < x < 2 or x > 3

Interval Notation

(1, 2) \cup (3, \infty)

Solution steps

\frac{1}{x - 1} - \frac{1}{x - 2} + \frac{1}{x - 3} > 0

Rewrite in standard form

\frac{x ^{2} - 4 x + 5}{( x - 1 ) ( x - 2 ) ( x - 3 )} > 0

Identify the intervals

1 < x < 2 or x > 3

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

\frac{( x ^{2} + 6 x + 4 )}{( 3 x ^{2} - 9 x + 4 )} \leq 0

(x - 2) (x + 7) < 0

(x - 1) - 7 > (x - 2)^{2}

6 x \cdot (- 5) x - 3 x^{2} - 2 x + 6 < 4

What is 1/((x-1))-1/((x-2))+1/((x-3))>0 ?
The solution to 1/((x-1))-1/((x-2))+1/((x-3))>0 is 1<x<2\lor x>3