What is (x^3+6x^2+11x+6)/(5x^3-12x^2+7x-6)>0 ?

The solution to (x^3+6x^2+11x+6)/(5x^3-12x^2+7x-6)>0 is x 2

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(x^3+6x^2+11x+6)/(5x^3-12x^2+7x-6)>0

\frac{x ^{3} + 6 x ^{2} + 11 x + 6}{5 x ^{3} - 12 x ^{2} + 7 x - 6} > 0

x < - 3 or - 2 < x < - 1 or x > 2

Interval Notation

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

Solution steps

\frac{x ^{3} + 6 x ^{2} + 11 x + 6}{5 x ^{3} - 12 x ^{2} + 7 x - 6} > 0

Factor

\frac{x ^{3} + 6 x ^{2} + 11 x + 6}{5 x ^{3} - 12 x ^{2} + 7 x - 6} : \frac{( x + 1 ) ( x + 2 ) ( x + 3 )}{( x - 2 ) ( 5 x ^{2} - 2 x + 3 )}

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

Identify the intervals

x < - 3 or - 2 < x < - 1 or x > 2

x^{\frac{1}{2}} \cdot x^{\frac{1}{3}}

- 2 x^{2} - 8 x - 6 < 0

4 x^{2} - 48 x + 144 = 0

4 (x + 1) = 2 (x + 5)

3 x < - 18.6

What is (x^3+6x^2+11x+6)/(5x^3-12x^2+7x-6)>0 ?
The solution to (x^3+6x^2+11x+6)/(5x^3-12x^2+7x-6)>0 is x<-3\lor-2<x<-1\lor x>2