Final answer:
To find the values of x for which (x²+x+3)/(2x²+x-6)≥0, we factor the denominator, find its zeros, and determine the signs of the intervals between these zeros. The numerator does not have real zeros, so it's always positive. The correct intervals for x are x≤-2 or x≥1.5.
Step-by-step explanation:
To find all values of x such that (x²+x+3)/(2x²+x-6)≥0, we need to consider the zeros of the numerator and the denominator, as these are the values for which the expression is undefined or equal to zero. We want to determine the signs of each interval between these zeros.
Firstly, factor the denominator (if possible) to find its zeros:
- 2x² + x - 6 = (2x - 3)(x + 2)
The zeros of the denominator are x = 1.5 and x = -2. Since the expression is undefined for these values, these are critical points that divide the number line into intervals.
Similarly, check if the numerator can be factored to find its zeros, but in this case, the numerator x² + x + 3 does not have real zeros as its discriminant (b² - 4ac) is negative.
Next, determine the sign of each interval obtained by the critical points. Note that the numerator is always positive since it has no real zeros. So the sign of the expression depends only on the sign of the denominator:
- For x < -2, the expression is positive since both terms (2x - 3) and (x + 2) are negative.
- For -2 < x < 1.5, the expression is negative because (2x - 3) is negative and (x + 2) is positive.
- For x > 1.5, the expression is positive as both terms are positive.
Combining these results gives us the intervals where the original expression is non-negative.
Therefore, the solution set for (x²+x+3)/(2x²+x-6)≥0 is x≤-2 or 1.5≤x.