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Math Help - Simple Interval finding?

  1. #1
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    Simple Interval finding?

    I've done problems exactly like these, but these two keep giving me the wrong answer, and I'm not sure why.

    1) if |x - 2| < delta then |1/x - 0.5| < 0.2
    2) if |x - 4| < delta then |sqrt(x) - 2| < 0.4

    In both I need to find delta.
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  2. #2
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    Quote Originally Posted by CFem View Post
    I've done problems exactly like these, but these two keep giving me the wrong answer, and I'm not sure why.

    1) if |x - 2| < delta then |1/x - 0.5| < 0.2
    2) if |x - 4| < delta then |sqrt(x) - 2| < 0.4

    In both I need to find delta.
    For 2.

    Try multiplying |\sqrt{x} - 2| by its conjugate.

    Notice that

    |\sqrt{x} -2| < 0.4

    |\sqrt{x} - 2| + 4 < 0.4 + 4

    |\sqrt{x} + 2| < 4.4.


    So

    |\sqrt{x} - 2||\sqrt{x} + 2| < 0.4\cdot 4.4

    |(\sqrt{x} - 2)(\sqrt{x} + 2)| < 1.76

    |x - 4| < 1.76.


    So \delta = 1.76.
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