So the question:
is 1/r > r/(r^2+2)?
The actual answer is, "we don't have enough info." according to the book, but can't we cross multiply, giving us:
r^2+2 > r^2
Which would be true in all cases, isn't it?
We supposedly would have enough info if we knew also knew that r>0, but what difference will that make? Squaring anything would make r^2 a positive number anyway, right?
Can anyone tell me what I'm missing here?
CaptainBlack gave one example of when this identity is not true (granted, his example was for r=-2).
As for your second question:
If we know that then what you did is ok. However, if , cross multiplying is illegal without also switching the inequality. In fact, because 1/r would be undefined.Hmm... Then did I do something wrong by cross multiplying the original equation in the first place?
Let me rephrase the question in another way: "Is always true? -- Answer: Yes/No" And the answer was "No.", as you guys explained that it would depend on whether r is a postive or a negative.
So I guess in cases where we don't know any information about the variable r (where it's positive or negative), cross multiplying to make a easier visual representation (where I made the equation ) of the problem would be ill advised. That's what I'm getting here. Is that about right?