Hi,

I am extremely confused about inequalities and would appreciate it if someone could help me out. I am not sure how to even phrase what is confusing me so I'll throw out a few example questions:

I suspect that for the 2nd and 3rd terms the factors would be (x+7) and (x-2), but I do not know what to do with the first term.

Another example:

In this case I have determined the following:

Code:

x < -2/3 and x < 3
and
x > -2/3 and x > 3

I do not know why I need those or what they mean, but my textbook said that I need to have both a positive and negative case, thus flipping the direction of the inequality sign. According to WolframAlpha's inequality solver the answer is: -2/3 < x < 3. I can sort of see how they determined the answer from my own findings, but what I do not understand is HOW or WHY they knew to choose x < 3 and not x > 3, and x > -2/3 instead of x < -2/3.

Thanks.