What is ((2x+1))/((x^2-x-12))<= 1/2 ?

The solution to ((2x+1))/((x^2-x-12)) = 7

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((2x+1))/((x^2-x-12))<= 1/2

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

x < - 3 or - 2 \leq x < 4 or x \geq 7

Interval Notation

(- \infty, - 3) \cup [- 2, 4) \cup [7, \infty)

Solution steps

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

Rewrite in standard form

\frac{- x ^{2} + 5 x + 14}{( x + 3 ) ( x - 4 )} \leq 0

Factor

\frac{- x ^{2} + 5 x + 14}{( x + 3 ) ( x - 4 )} : \frac{- ( x + 2 ) ( x - 7 )}{( x + 3 ) ( x - 4 )}

\frac{- ( x + 2 ) ( x - 7 )}{( x + 3 ) ( x - 4 )} \leq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x + 2 ) ( x - 7 ) ) ( - 1 )}{( x + 3 ) ( x - 4 )} \geq 0 \cdot (- 1)

Simplify

\frac{( x + 2 ) ( x - 7 )}{( x + 3 ) ( x - 4 )} \geq 0

Identify the intervals

x < - 3 or - 2 \leq x < 4 or x \geq 7

(x + 9) (x - 3) > 0

(x - 1)^{2} - 11 < (x - 2)^{2}

4 x^{2} + 2 > 0

x^{2} \geq \frac{1}{2}

x (x + 5)^{2} (x - 3) < 0

What is ((2x+1))/((x^2-x-12))<= 1/2 ?
The solution to ((2x+1))/((x^2-x-12))<= 1/2 is x<-3\lor-2<= x<4\lor x>= 7