# Inequality question:

• May 28th 2007, 11:25 AM
thumpin_termis
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? :confused:
• May 28th 2007, 11:29 AM
CaptainBlack
Quote:

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? :confused:

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
• May 28th 2007, 11:47 AM
thumpin_termis
Hmm... Then did I do something wrong by cross multiplying the original equation in the first place?
• May 28th 2007, 12:37 PM
Plato
Quote:

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.
• May 28th 2007, 12:47 PM
ecMathGeek
Quote:

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? :confused:

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).

Quote:

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.
• May 28th 2007, 01:33 PM
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.

Quote:

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?
• May 28th 2007, 01:36 PM
CaptainBlack
Quote:

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