What is (((-18x^2+18))/((x^2+3)^3))>0 ?

The solution to (((-18x^2+18))/((x^2+3)^3))>0 is -1<x<1

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(((-18x^2+18))/((x^2+3)^3))>0

(\frac{( - 18 x ^{2} + 18 )}{( x ^{2} + 3 ) ^{3}}) > 0

- 1 < x < 1

Interval Notation

(- 1, 1)

Solution steps

\frac{- 18 x ^{2} + 18}{( x ^{2} + 3 ) ^{3}} > 0

Factor

\frac{- 18 x ^{2} + 18}{( x ^{2} + 3 ) ^{3}} : \frac{- 18 ( x + 1 ) ( x - 1 )}{( x ^{2} + 3 ) ^{3}}

\frac{- 18 ( x + 1 ) ( x - 1 )}{( x ^{2} + 3 ) ^{3}} > 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 18 ( x + 1 ) ( x - 1 ) ) ( - 1 )}{( x ^{2} + 3 ) ^{3}} < 0 \cdot (- 1)

Simplify

\frac{18 ( x + 1 ) ( x - 1 )}{( x ^{2} + 3 ) ^{3}} < 0

Divide both sides by

18

\frac{\frac{18 ( x + 1 ) ( x - 1 )}{( x ^{2} + 3 ) ^{3}}}{18} < \frac{0}{18}

Simplify

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

Identify the intervals

- 1 < x < 1

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

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

\frac{( x ^{2} + 6 x - 11 )}{( x + 3 )} < - 1

(4 x - 5)^{2} - 16 x (x + 3) < - 21

\frac{( 2 x + 1 )}{( x ^{2} - x - 12 )} \leq \frac{1}{2}

What is (((-18x^2+18))/((x^2+3)^3))>0 ?
The solution to (((-18x^2+18))/((x^2+3)^3))>0 is -1<x<1