What is ((6x-x^2))/(((x+9)^2))<= 0 ?

The solution to ((6x-x^2))/(((x+9)^2)) = 6

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((6x-x^2))/(((x+9)^2))<= 0

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

x < - 9 or - 9 < x \leq 0 or x \geq 6

Interval Notation

(- \infty, - 9) \cup (- 9, 0] \cup [6, \infty)

Solution steps

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

Factor

\frac{6 x - x ^{2}}{( x + 9 ) ^{2}} : \frac{- x ( x - 6 )}{( x + 9 ) ^{2}}

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

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - x ( x - 6 ) ) ( - 1 )}{( x + 9 ) ^{2}} \geq 0 \cdot (- 1)

Simplify

\frac{x ( x - 6 )}{( x + 9 ) ^{2}} \geq 0

Identify the intervals

x < - 9 or - 9 < x \leq 0 or x \geq 6

\frac{2}{3 x} > - 6 x

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

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

\frac{( 2 x + 1 )}{( 3 x - 6 )} \geq 3

100 - x^{2} \geq 0

What is ((6x-x^2))/(((x+9)^2))<= 0 ?
The solution to ((6x-x^2))/(((x+9)^2))<= 0 is x<-9\lor-9<x<= 0\lor x>= 6