SOLUTION: Solve the rational inequality. Express answer using interval notation (show work) x^2-x-12 _________>0 x^2+x-6

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Question 1164558: Solve the rational inequality. Express answer using interval notation (show work)
x^2-x-12
_________>0
x^2+x-6
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
Solve the rational inequality. Express answer using interval notation (show work)
x^2-x-12
_________>0
x^2+x-6
~~~~~~~~~~~~~~ 


The numerator   x^2-x-12 = (x-4)*(x+3).

The denominator x^2+x-6  = (x-2)*(x+3).


The denominator is zero at x= 3;  so this value is excluded from the domain.

The numerator   is zero at x= 4  and x=-3;  so these values are excluded from the solution set.



After canceling the common factor (x+3) in the numerator and denominator, you get the function in the form  .


It is greater than 0, when linear binomials are EITHER both negative OR both positive. 


So the solution set is the union of intervals  {x < -3} U (-3 < x < 2} U {x > 4},   or    (-oo,-3) U (-3,2) U (4,oo).      ANSWER>


See the plot below


    


    Plot y = 



Notice that the point x = -3 is the  in the domain of the function: formally, the function IS NOT DEFINED at this point.

Solved.


Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your inequality is:

(x^2 - x - 12)/(x^2 + x - 6) > 0

factor these quadratics to get:

(x-4)*(x+3)/((x-2)*(x+3)) > 0

the (x+3) in the numerator and the denominator cancel out and you are left with:

(x-4)/(x-2) > 0

this will be positive when x is greater than 4 or when x is smaller than 2

it till be negative when x is > 2 and < 4.

when x is greater than 4, you have a positive numerator divided by a positive denominator which results in a positive number.

when x is less than 2, you have a negative numerator divided by a negative denominator which results in a positive number.

when x is greater than 2 and less than 4, you have a negative numerator divided by a positive denominator which results in a negative number.

this can be seen in the following graph.



the shaded area is when the resulting number is positive.
the non-shaded area is when the resulting number if negative.


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