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Math Help - Use of modulus..

  1. #1
    Newbie findmehere.genius's Avatar
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    Use of modulus..

    I have tried a lot of questions on Modulus...yet it creates trouble for me when it comes to what would be the answer to a questions like these:
    ||x-2|-2|=x find value of x
    and one more, |3-|2x+5||=x+2 find x..


    answer to the first one is 0<=x<=2 and of second one is 0, -4/3

    I will be thankful to any sort of help.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by findmehere.genius View Post
    I have tried a lot of questions on Modulus...yet it creates trouble for me when it comes to what would be the answer to a questions like these:
    ||x-2|-2|=x find value of x
    and one more, |3-|2x+5||=x+2 find x..


    answer to the first one is 0<=x<=2 and of second one is 0, -4/3

    I will be thankful to any sort of help.
    Hi

    ||x-2|-2|=x find value of x

    Since there is |x-2| you need to consider 2 cases :
    1) x\leq 2
    In this case |x-2| = 2-x (because x-2 \leq 0)
    The equation becomes |-x|=x or |x|=x (because |-x|=|x|)
    which means x\geq 0
    The solutions in this case are therefore 0 \leq x \leq 2

    2) x\geq 2
    In this case |x-2| = x-2 (because x-2 \geq 0)
    The equation becomes |x-4|=x

    2 sub-cases :
    2a) x\leq 4
    The equation becomes 4-x=x which means x=2
    2b) x\geq 4
    The equation becomes x-4=x which has no solution
    The solutions in this case are therefore x = 2

    The overall solutions are therefore 0 \leq x \leq 2
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  3. #3
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    The method is the same for the second one

    |3-|2x+5||=x+2 find x

    Case 1 : x \leq -\frac52
    The equation becomes |3-(-2x-5)|=x+2 or |2x+8|=x+2

    Case 1a : x \leq -4
    The equation becomes -(2x+8)=x+2 or x = -\frac{10}{3}
    But -\frac{10}{3} > -4 therefore there is no solution for this case

    Case 1b : x \geq -4
    The equation becomes 2x+8=x+2 or x = -6
    But -6 < -4 therefore there is no solution for this case

    Case 2 : x \geq -\frac52
    The equation becomes |3-(2x+5)|=x+2 or |-2x-2|=x+2

    Case 2a : x \leq -1
    The equation becomes -2x-2=x+2 or x = -\frac43
    And -\frac43 \leq -1 therefore -\frac43 is one solution

    Case 2b : x \geq -1
    The equation becomes 2x+2=x+2 or x = 0
    And 0 \geq -1 therefore 0 is one solution

    Finally the set of solutions is -4/3 and 0
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  4. #4
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    Hello, findmehere.genius!

    I graphed the first one . . . Got the same answer as running-gag.


    \bigg||x-2|-2\bigg|\:=\:x . Find value of x.
    I'll graph it in baby steps . . .

    We know what y \:=\:x looks like:
    Code:
                    |
                    |     *
                    |   *
                    | *
          - - - - - * - - - - -
                  * | 
                *   |
              *     |
                    |

    Then we have: . y \:=\:|x|
    The absolute value says: anything below the x-axis is reflected upward.
    Code:
                    |
            *       |       *
              *     |     *
                *   |   *
                  * | *
          - - - - - * - - - - -
                    |
                    |

    Then y \:=\:|x-2| moves the graph 2 units to the right.
    Code:
            *   |           *
              * |         *
                *       *
                | *   *
          - - - + - * - - - - -
                |   2
                |

    Then y \:=\:|x-2| -2 moves the graph down 2 units.
    Code:
                |
          *     |             *
            *   |           *
              * |         *
          - - - * - - - * - - -
                | *   *
                |   *
                |

    Finally, y \:=\:||x-2| - 2|
    . . Anything below the x-axis is reflected upward.
    Code:
                |
          *     |             *
            *   |   *       *
              * | *   *   *
          - - - * - + - * - - - -
                |   2
                |
    There!


    What does this graph have in common with y = x ?
    . . (See the first graph.)


    We find that the graphs coincide for: .  0 \:\leq\:x\:\leq\:2

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