1. ## Inequality question:

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?

2. Originally Posted by thumpin_termis
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?
Suppose $r=-2$ is the inequality now satisfied?

The righthand side is $-1/2$ and the left hand side
is $-1/3$, but $-1/2<-1/3$

RonL

3. Hmm... Then did I do something wrong by cross multiplying the original equation in the first place?

4. Originally Posted by thumpin_termis
Then did I do something wrong by cross multiplying the original equation in the first place?
What exactly is the question?
As given, the inequality does indeed have a solution.
It is true for all positive real numbers.
It is false for all negative real numbers.

5. Originally Posted by thumpin_termis
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?
I'm confused by your question. Are you asking if we are able to find a solution to this problem? Or are you asking if this identity is always true?

CaptainBlack gave one example of when this identity is not true (granted, his example was for r=-2).

Hmm... Then did I do something wrong by cross multiplying the original equation in the first place?
If we know that $r>0$ then what you did is ok. However, if $r <0$, cross multiplying is illegal without also switching the inequality. In fact, $r \not= 0$ because 1/r would be undefined.

6. Let me rephrase the question in another way: "Is $1/r > r/(r^2+2)$ 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.

Originally Posted by ecMathGeek

If we know that $r>0$ then what you did is ok. However, if $r <0$, cross multiplying is illegal without also switching the inequality. In fact, $r \not= 0$ because 1/r would be undefined.
That's the part I was confused about.

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 $r^2+2 > r^2$) of the problem would be ill advised. That's what I'm getting here. Is that about right?

7. Originally Posted by thumpin_termis
Let me rephrase the question in another way: "Is $1/r > r/(r^2+2)$ 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.

That's the part I was confused about.

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 $r^2+2 > r^2$) of the problem would be ill advised. That's what I'm getting here. Is that about right?
Yes, it would be wrong, since you may be effectivly multiplying through
by a negative number which would revese the inequality.

RonL