SOLUTION: Solve the rational inequality. Express answer using interval notation (show work) x^2-x-12 ________

Click here to see ALL problems on Equations

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.

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