SOLUTION: For what values of x is {x^2 + x + 3}/{2x^2 + x - 6} >= 0? Note: Be thorough and explain why all points in your answer are solutions and why all points outside your answer are

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: For what values of x is {x^2 + x + 3}/{2x^2 + x - 6} >= 0? Note: Be thorough and explain why all points in your answer are solutions and why all points outside your answer are       Log On


   

Question 1025840: For what values of x is
{x^2 + x + 3}/{2x^2 + x - 6} >= 0?
Note: Be thorough and explain why all points in your answer are solutions and why all points outside your answer are not solutions.
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the critical points of the functions.
x%5E2%2Bx%2B3=0
The roots of this quadratic are complex so no real roots.
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2x%5E2%2Bx-6=0
%28x%2B2%29%282x-3%29=0
x%2B2=0
x=-2
and
2x-3=0
2x=3
x=3%2F2
Break up the number line using the two critical points.
Region 1 : (-infinity,-2)
Region 2 : (-2,3%2F2)
Region 3 : (3%2F2,infinity)
In each region, choose a point (not an endpoint) check the quotient.
Region 1 : x=-3
%28%28-3%29%5E2-3%2B3%29%2F%282%28-3%29%5E2-3-6%29=%289%29%2F%289%29=1
Region 2 : x=0
%28%280%29%5E2%2B0%2B3%29%2F%282%280%29%5E2%2B0-6%29=%283%29%2F%28-6%29=-1%2F2
Region 3 : x=2
%28%282%29%5E2%2B2%2B3%29%2F%282%282%29%5E2%2B2-6%29=%289%29%2F%284%29
So when,
(-infinity+%3Cx%3C+-2) ,f%28x%29%3E0
(-2+%3Cx%3C+3%2F2) ,f%28x%29%3C0
(3%2F2+%3Cx%3C+infinity) ,f%28x%29%3E0
The quotient is never equal to zero since the numerator is never equal to zero.
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Answer by ikleyn(52848) About Me  (Show Source):