For how many positive integers n

Question

For how many positive integers n < 100 is (n - 5)(n - 23)(n - 68) positive?

kinggeorge · Answer

To allow it to be positive, we need one of two cases. 

Case 1: All three of $(n-5),(n-23),(n-68)$ must be positive. 

Case 2: Only a single one is positive, and the other two are negative. 

Let's evaluate these one at a time.

anonymous · Answer

So It's positive if:
$$n > 68 \space or \space 6 < n < 23$$which is 48?

kinggeorge · Answer

Perfect. Here's the detailed explanation I wrote up.

Case 1: $n$ must be greater than 68, and less than 100. There are exactly $100-69=31$ possibilities. 

Case 2: $5<n<23$ Here, there are exactly $23-6=17$ possibilities. Add them together, we get$$31+17=48$$choices for $n$ so that $$(n-5)(n-23)(n-68)>0$$

anonymous · Answer

Oh ok. So it's very similar to quadratic inequalities right?

kinggeorge · Answer

Yes, I would call this similar to quadratic inequalities.

anonymous · Answer

Alright. Thank you!

kinggeorge · Answer

You're welcome.