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Math Help - Some Summer Calculus Homework Help...

  1. #1
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    Thumbs up Some Summer Calculus Homework Help...

    Okay I have two questions that involve absolute value and inequalities that I am having some difficulty figuring/solving out....

    Here they are,

    Problem #1: Show that if |a-b|<s and |c-b|<t, then |a-c|<s+t



    Next problem is a similar one to #1.


    Problem #2: Show that if |x-1|<(e/3) , then |(3x+2)-5|<e



    Any and all help is appreciated, thanks!
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by forkball42 View Post
    Problem #2: Show that if |x-1|<(e/3) , then |(3x+2)-5|<e
    |x - 1| < \frac{e}{3}

    3|x - 1| < e

    |3x - 3| < e

    |3x + (2 - 5)| < e

    |(3x + 2) - 5| < e

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by forkball42 View Post
    Problem #1: Show that if |a-b|<s and |c-b|<t, then |a-c|<s+t
    |a - b| < s
    and
    |c - b| < t

    Thus adding the two inequalities:
    |a - b| + |c - b| < s + t

    Now, the "triangle inequality" (which is equivalent to the "Cauchy-Schwartz inequality" in this case) says that:
    |a - c| \leq |a - b| + |c - b| <-- It is slightly easier to see that |a - c| \leq |a - b| + |b - c|, but as |c - b| = |b - c| these are the same.

    Thus
    |a - c| < s + t

    -Dan
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