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For what values of x is\( \frac{x^2 + x + 3}{2x^2 + x - 6} \ge 0? \)

Note: Be thorough and explain why all points in your answer are solutions and why all points outside your answer are not solutions.

 Jan 20, 2019

this means they are either both positive, both negative, or x^2+x+3 is 0.


x^2+x+3 can not be 0 because of the quadratic formula, so they are either both positive or both negative.


if they are both positive, then 2x^2+x-6 should be positive. factoring, it is (2x-3)(x+2), so the roots are 3/2 and -2.


since this is positive, x is greater than 3/2 or less than -2. in these cases, x^2+x+3 will be positive (plug in some numbers).


now you need to do the case where they are both negative.


in this case, x is between -2 and 3/2 in the denominator.


now you try plugging in values. the +3 in the numerator will overpower any value in there so it will not be negative, so we can rule this case out.


so the answer is \(\boxed{x<-2}\) or \(\boxed{x>{3\over2}}\).




 Jan 20, 2019

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