What is (((x-1)(3-x)))/((x^2-5x))>0 ?

The solution to (((x-1)(3-x)))/((x^2-5x))>0 is 0<x<1\lor 3<x<5

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(((x-1)(3-x)))/((x^2-5x))>0

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

0 < x < 1 or 3 < x < 5

Interval Notation

(0, 1) \cup (3, 5)

Solution steps

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

Factor

\frac{( x - 1 ) ( 3 - x )}{x ^{2} - 5 x} : \frac{( x - 1 ) ( 3 - x )}{x ( x - 5 )}

\frac{( x - 1 ) ( 3 - x )}{x ( x - 5 )} > 0

Identify the intervals

0 < x < 1 or 3 < x < 5

2 x^{2} - 3 x + 5 < 0

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

\frac{( x + 4 )}{( x - 2 )} < \frac{2}{( x + 1 )}

(x - 2)^{3} > 0

\frac{( ( x - 1 ) ^{2} \cdot ( x ^{2} - 5 ) )}{( x ^{2} + x - 1 )} \geq 0

What is (((x-1)(3-x)))/((x^2-5x))>0 ?
The solution to (((x-1)(3-x)))/((x^2-5x))>0 is 0<x<1\lor 3<x<5