Question 1164558
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Solve the rational inequality. Express answer using interval notation (show work)
x^2-x-12
_________>0
x^2+x-6
~~~~~~~~~~~~~~



<pre>
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  {{{(x-4)/(x-2)}}}.


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).      <U>ANSWER</U>>


See the plot below


    {{{graph( 400, 400, -10, 10, -10, 10,
              (x^2-x-12)/(x^2+x-6)
)}}}


    Plot y = {{{(x^2-x-12)/(x^2+x-6)}}}



Notice that the point x = -3 is the {{{highlight(hole)}}} in the domain of the function: formally, the function IS NOT DEFINED at this point.
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Solved.