express x^2-x-6>0 as an interval

Question

I want to know how to know what the interval is after i factor the equation. 

Answer 1

x²-x-6>0

The factors of -6 that add to -1 are (-3)(2) so this factors to

(x-3)(x+2)>0

 

Note:  This will be equal to zero for x=3 or x=-2

 

In order for this inequality to be greater than zero

either both x-3 and x+2 are positive

or both x-3 and x+2  are negative...

 

1.  x-3>0   .... x>3

    x+2>0 ....  x>-2

The interval where both are true is for all x>3

 

2.  x-3<0  ...   x<3

     x+2<0 ...   x<-2

The interval where both are true is for all x<-2

 

Your answer is the either  x>3   or   x<-2

 

In interval notation     (-∞,-2) U (3,∞)

 

Hope this helps.  Let me know if you have questions

Answer 2

First we can find the zeros.

 

(x - 3)(x + 2) = 0

 

x = -2        and          x = 3

 

These are our zeros.  Next, we use test points to evaluate the left side of the inequality.  Lets choose x=-3 , x=0 , and x=4.  We want the result to be positive.

 

For x=-3:

 

(-3 - 3)(-3 + 2) = (-6)(-1) = 6

 

 

For x=0:

 

The result is -6.

 

 

For x=4:

 

(4 - 3)(4 + 2) = (1)(6) = 6

 

 

The x values that make the result positive are in the interval  (-∞, -2)∪(3, ∞).