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.
+532 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 \(\boxed{x>{3\over2}}\).
HOPE THIS HELPED!!
