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

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) (Show Source): You can put this solution on YOUR website!
Find the critical points of the functions.

The roots of this quadratic are complex so no real roots.
.
.

and

Break up the number line using the two critical points.
Region 1 : (,)
Region 2 : (,)
Region 3 : (,)
In each region, choose a point (not an endpoint) check the quotient.
Region 1 :

Region 2 :

Region 3 :

So when,
() ,
() ,
() ,
The quotient is never equal to zero since the numerator is never equal to zero.
.
.
.
.
Answer by ikleyn(52851) (Show Source): You can put this solution on YOUR website!
.
See also the lesson
Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials
in this site, where you can find similar solved problems.

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