What is (2x-x^2)/((3+x)*(5-x))<0 ?

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

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(2x-x^2)/((3+x)*(5-x))<0

\frac{2 x - x ^{2}}{( 3 + x ) \cdot ( 5 - x )} < 0

- 3 < x < 0 or 2 < x < 5

Interval Notation

(- 3, 0) \cup (2, 5)

Solution steps

\frac{2 x - x ^{2}}{( 3 + x ) ( 5 - x )} < 0

Factor

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

\frac{- x ( x - 2 )}{( 3 + x ) ( 5 - x )} < 0

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

- 3 < x < 0 or 2 < x < 5

x^{2} \cdot (x + 2) < 0

\frac{( x - 12 ) ( x - 4 )}{( x ^{2} - 16 )} > 0

\frac{( 12 x )}{( x + 2 )} < 1

(x + 4) (x - 8) > 3 (x - 8)

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

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