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Math Help - Related range of values of f(x) necessary/sufficient/ with absolute values

  1. #1
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    Thumbs up Related range of values of f(x) necessary/sufficient/ with absolute values

    Hi Forum.

    I've asked a lot of times about absolute values.
    But here I go again:

    Consider two conditions
    x^2-3x-10 and |x-2|<a

    Where x is a real number, and a is a positive real number.

    The range of a, so that |x-2|<a is a necessary condition for x^2-3x-10<0

    The range of a, so that |x-2|<a is a sufficient condition for x^2-3x-10<0

    This problem is mainly about logic, right?
    Here is something that I thought, after calculating the range of x^2-3x-10<0, that is -2<x<5

    If we have -2<x<5
    -2________________________________5
    |________________________________|
    ___________________________a=0 ______________________________ a=3
    |_________________________________________________ _| <-|x-2|<a


    Edit: This didn't work so well but, the a=0 is right below -2 and the a=3 is right below 5


    This is our sufficient condition: 0<a<3
    I'm not sure what the absolute value does here.
    What does the absolute value mean?
    Absolute values are only the distance to x=0, right?

    After calculating -correctly- you get a nicer solution that is:
    0<a\leqslant  3

    But how do we get such solution?

    |x-2|<a is
    x-2 if x-2\geqslant  a and
    -x+2 if  x-2<a

    This relates to two range of values so this got a little out of control.
    Somehow confusing, I wish I knew somewhere to find exercises of this kind!

    I'm confused about that necessary/sufficient part

    Sufficient
    If you mow the lawn, you receive 10 dollars.
    ____________________Necessary

    The 0<a\leqslant  3 range is sufficient for f(x) to work, or something like that?
    But where does necessity comes in this? I know that if something is necessary it is not sufficient.

    A--->B
    A implies B
    A sufficient, B necessary

    I appreciate any help!
    Thanks!
    Last edited by Zellator; May 31st 2011 at 08:50 PM. Reason: Correction
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  2. #2
    A Plied Mathematician
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    Interesting problem. Your problem is dealing with implication. Let's let P be the proposition that

    x^{2}-3x-10<0,

    and let's let Q(a) be the proposition that

    |x-2|<a.

    Now, if you want to find the range of a such that Q(a) is sufficient for P, then that's the same thing as finding the range of a such that Q(a) implies P.

    Conversely (literally!), if you want to find the range of a such that Q(a) is necessary for P, then that's the same thing as finding the range of a such that P implies Q(a). So here, you just reverse the direction of the implication.

    I agree with your 'sufficient' answer. 0<a\le 3 is exactly correct.

    But now, with the 'necessary' answer, you want P to imply Q(a). How could that happen?
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