I know you have answered this question before. Sorry to bother you:(
But is there any other way besides using graphing calculator?
For how many integers n is (2n+1)(3n-11) a negative number?
Of course there is. (In the future, please ask this as a comment on the post where I talked about this question before. That way all the conversation about the question can be in the same place!)
(2n+1)(3n-11) will be negative when one of the factors is negative and the other is positive. When they’re both positive, it’ll be positive. When they’re both negative, it’ll also be positive. So figure out how many values of n will make one factor negative and the other one positive. (It’s also helpful to know that the graph of this will make a parabola, so it’ll only be negative for a while: once it goes from positive to negative to positive once, it won’t do it again.)
The key to doing that efficiently is recognizing that when n is negative, the expression will always be positive, so you only need to try zero and a few positive numbers.
(2(0) + 1)(3(0) – 11) = -11
(2(1) + 1)(3(1) – 11) = -24
(2(2) + 1)(3(2) – 11) = -25
(2(3) + 1)(3(3) – 11) = -14
(2(4) + 1)(3(4) – 11) = 9