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Math Help - Inequality question:

  1. #1
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    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?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by thumpin_termis View Post
    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
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  3. #3
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    Hmm... Then did I do something wrong by cross multiplying the original equation in the first place?
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  4. #4
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    Quote Originally Posted by thumpin_termis View Post
    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.
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  5. #5
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by thumpin_termis View Post
    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).

    As for your second question:

    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.
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  6. #6
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    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 View Post
    As for your second question:

    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?
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by thumpin_termis View Post
    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
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