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Interval Notation

  1. Jan 3, 2006 #1
    A friend had a question about Interval Notations, I had no idea how to do it..but kind of ..well tried to learn how. And I think the answer is A? Any help here?

    1: Write in interval notation: | x | ≤ 3
    a. (-∞, -3] U [3, ∞)
    b. (-∞, ∞)
    c. [-3, 3]
    d. No solution
     
  2. jcsd
  3. Jan 3, 2006 #2
    With regards to interval notation,
    ( or ) correspond to "up to but not including",
    [ or ] correspond to "up and and including",
    and U corresponds to "union", or "and this stuff too".

    So, for instance, option A would read as:

    everything from negative infinity (although not including negative infinity, since it technically isn't a number and can never be reached) up to and including negative three, in addition to everything from (and including) positive three up to (but not including) positive infinity.

    An easy way to check whether or not that matches the condition you want is to pick a few numbers from your interval and see if they satisfy the condition. If you can find any numbers which do not satisfy the condition then the interval won't work.
     
  4. Jan 3, 2006 #3

    Tx

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    From my knowledge of Absolute Values it means that there is two seperate solutions, so it can be either -x or x. So Seeing that this has to be less than or equal to 3, I would go with option A.
     
    Last edited by a moderator: Jan 3, 2006
  5. Jan 3, 2006 #4
    Thank you for the quick help. I figured it was A because of the absolute value...
     
  6. Jan 3, 2006 #5

    George Jones

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    Staff Emeritus
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    If [tex]x = -10[/tex], is it true that [tex]|x| \leq 3[/tex]?

    Regards,
    George
     
  7. Jan 3, 2006 #6
    Umm... To me that equation says that the absolute value of x must be less than or equal to 3 not greater than or equal to.
     
  8. Jan 3, 2006 #7

    Tx

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    Yep, Sorry about that.
     
  9. Jan 7, 2006 #8
    The absolute value of x is a piecewise defined function therefore we have,

    [tex]
    \left| x \right| \le 3\left\{ \begin{array}{l}
    x \le 3,{\rm when \ x } \ge {\rm 0} \\
    - x \le 3,{\rm when \ x < 0} \\
    \end{array} \right.
    [/tex]

    Solving for x when [tex]{\rm x} \ge {\rm 0}[/tex] we have,

    [tex]
    x \le 3
    \]
    [/tex]

    Solving for x when [tex]{\rm x < 0}[/tex] we have,

    [tex]
    x \ge - 3

    [/tex]

    We conclude that the domain of x is [tex][ - 3,3] \ \ \ \ \ \ \ \square [/tex]
     
    Last edited: Jan 7, 2006
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