What is ((-x^2+9x-14))/((2x^2+9))>= 0 ?

The solution to ((-x^2+9x-14))/((2x^2+9))>= 0 is 2<= x<= 7

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((-x^2+9x-14))/((2x^2+9))>= 0

\frac{( - x ^{2} + 9 x - 14 )}{( 2 x ^{2} + 9 )} \geq 0

2 \leq x \leq 7

Interval Notation

[2, 7]

Solution steps

\frac{- x ^{2} + 9 x - 14}{2 x ^{2} + 9} \geq 0

Factor

\frac{- x ^{2} + 9 x - 14}{2 x ^{2} + 9} : \frac{- ( x - 2 ) ( x - 7 )}{2 x ^{2} + 9}

\frac{- ( x - 2 ) ( x - 7 )}{2 x ^{2} + 9} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - 2 ) ( x - 7 ) ) ( - 1 )}{2 x ^{2} + 9} \leq 0 \cdot (- 1)

Simplify

\frac{( x - 2 ) ( x - 7 )}{2 x ^{2} + 9} \leq 0

Identify the intervals

2 \leq x \leq 7

x - > \frac{- 2 ( 8 + x ^{3} )}{( x ^{2} - 4 )}

(x + 2) \cdot (x - 3) > (2 - x) \cdot (6 - x)

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

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

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

What is ((-x^2+9x-14))/((2x^2+9))>= 0 ?
The solution to ((-x^2+9x-14))/((2x^2+9))>= 0 is 2<= x<= 7