What is ((x^2-6x+9))/((5-4x-x^2))>= 0 ?

The solution to ((x^2-6x+9))/((5-4x-x^2))>= 0 is -5<x<1\lor x=3

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((x^2-6x+9))/((5-4x-x^2))>= 0

\frac{( x ^{2} - 6 x + 9 )}{( 5 - 4 x - x ^{2} )} \geq 0

- 5 < x < 1 or x = 3

Interval Notation

(- 5, 1) \cup x = 3

Solution steps

\frac{x ^{2} - 6 x + 9}{5 - 4 x - x ^{2}} \geq 0

Factor

\frac{x ^{2} - 6 x + 9}{5 - 4 x - x ^{2}} : \frac{( x - 3 ) ^{2}}{- ( x - 1 ) ( x + 5 )}

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

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

- 5 < x < 1 or x = 3

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

x, f < 0

(x - 1)^{2} \cdot (x + 1)^{3} \cdot (x - 4) < 0

2 - x - x^{2} \geq 0

\frac{( x + 2 )}{( x - 2 )} \leq 2

What is ((x^2-6x+9))/((5-4x-x^2))>= 0 ?
The solution to ((x^2-6x+9))/((5-4x-x^2))>= 0 is -5<x<1\lor x=3