#1**+1 **

This one is a little difficult.....however.......note that x^2 + x + 3 is positive for all x

We can see this because this is a parabola that turns upward and the vertex is (-1/2, 2.75) .....which is above the x axis......so this part of the function is never negative

So.....only the denominator can make this function negative....and we have

[ 2x^2 + x - 6 ] ≥ 0

Factor the denominator

(2x -3) ( x + 2) ≥ 0

Setting each factor to 0 and solving for x, we have that x = 3/2 and x = -2

And we can see that when -2 < x < 3/2 the denominator will be negative

Then...the function will be ≥ 0 on these intervals :

(-infinity , -2) U (3/2, +infinity)

Notice that we have to exclude -2 and 3/2 from the soution because these would make the denominator 0

See the graph here : https://www.desmos.com/calculator/dztw4t5o0m

CPhill
Apr 6, 2017