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Math Help - Find all possible values for the inequality.

  1. #1
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    Find all possible values for the inequality.

    Question: What are the possible values of |2x3| when 0<|x1|<2?




    My solution:

    We know
    x1
    becomes x-1 if x-1≥0 and -(x-1) if x-1<0.
    Now consider two cases.
    Case 1:
    0<x-1<2
    1<x<3
    -1<2x-3<3.

    Case 2:
    0<-x+1<2
    -1<x<1
    -2<2x<2
    -5<2x-3<-1

    Then the possible value include |2x-3|<3. Am I solving this problem correctly ??
    Note, -5<|2x-3|<-1 would not work since the |2x-3| is bounded between two negative values.

    Am I solving this problem correctly ?? The solution seems so incomplete?
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  2. #2
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    Re: Find all possible values for the inequality.

    If 0< |x- 2|< 2 then -2< x- 2< 2 (with x- 2\ne 0). Adding 2 to each part, 0< x< 4 (with x\ne 2). Multiplying by 2, 0< 2x< 8 (with 2x\ne 4). Subtracting 3, -3< 2x- 3< 5 (with 2x- 3\ne 1). That is, if 0< |x- 2|< 2, 2x- 3 can be any number strictly between -3 and 5, except 1.
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  3. #3
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    Re: Find all possible values for the inequality.

    Hey turbozz.

    Take a look at this thread:

    Calculus Inequality Conundrum
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  4. #4
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    Re: Find all possible values for the inequality.

    Hello, turbozz!

    \text{What are the possible values of }|2x-3|\,\text{ when }\,0 < |x-1| < 2\,?

    \begin{array}{ccccccc}\text{Given: } & 0 & < & |x-1| & < & 2 \\ \\ \text{Then:} & \text{-}2 & < & x-1 & < & 2 \\ \\ \text{Add 1:} & \text{-}1 & < & x & < & 3 \\ \\  \text{Times 2} & \text{-}2 & < & 2x & < & 6 \\ \\ \text{Minus 3:} & \text{-}5 &<& 2x-3 &<& 3  \end{array}
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  5. #5
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by HallsofIvy View Post
    If 0< |x- 2|< 2 then -2< x- 2< 2 (with x- 2\ne 0). Adding 2 to each part, 0< x< 4 (with x\ne 2). Multiplying by 2, 0< 2x< 8 (with 2x\ne 4). Subtracting 3, -3< 2x- 3< 5 (with 2x- 3\ne 1). That is, if 0< |x- 2|< 2, 2x- 3 can be any number strictly between -3 and 5, except 1.
    I don't understand why your using 0<|x-2|<2 ?? How does this relate to the inequality at hand?
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  6. #6
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by Soroban View Post
    Hello, turbozz!


    \begin{array}{ccccccc}\text{Given: } & 0 & < & |x-1| & < & 2 \\ \\ \text{Then:} & \text{-}2 & < & x-1 & < & 2 \\ \\ \text{Add 1:} & \text{-}1 & < & x & < & 3 \\ \\  \text{Times 2} & \text{-}2 & < & 2x & < & 6 \\ \\ \text{Minus 3:} & \text{-}5 &<& 2x-3 &<& 3  \end{array}
    how to do go from 0 <|x-1|<2 to -2<x-1<2 ? And x=1 wouldn't work since it wouldn't satisfy 0<|x-1|<2.
    Last edited by turbozz; September 29th 2013 at 05:39 PM.
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  7. #7
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by turbozz View Post
    Question: What are the possible values of |2x3| when 0<|x1|<2?




    My solution:

    We know
    x1
    becomes x-1 if x-1≥0 and -(x-1) if x-1<0.
    Now consider two cases.
    Case 1:
    0<x-1<2
    1<x<3
    -1<2x-3<3.

    Case 2:
    0<-x+1<2
    -1<x<1
    -2<2x<2
    -5<2x-3<-1

    Then the possible value include |2x-3|<3. Am I solving this problem correctly ??
    Note, -5<|2x-3|<-1 would not work since the |2x-3| is bounded between two negative values.

    Am I solving this problem correctly ?? The solution seems so incomplete?
    The triangle inequality works nicely here to get an upper bound: \displaystyle \begin{align*} |a + b| \leq |a| + |b| \end{align*}

    So \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\leq |2x - 2| + |-1| \\ &= 2|x - 1| + 1 \\ &< 2(2) + 1 \textrm{ since } |x - 1| < 2 \\ &= 5 \end{align*}

    So we can say \displaystyle \begin{align*} |2x - 3| < 5 \end{align*}.

    Also, the reverse triangle inequality is useful here to get a lower bound. \displaystyle \begin{align*} \left| |a| - |b| \right| \leq |a - b| \end{align*}. So that means

    \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\geq \left| |2x - 2| - |1| \right| \textrm{ by the Reverse Triangle Inequality} \\ &= \left| 2|x - 1| - 1 \right| \\ &> \left| 2(0) - 1 \right| \textrm{ since } |x - 1| > 0 \\ &= |-1| \\ &= 1  \end{align*}

    So that means we have \displaystyle \begin{align*} 1 < |2x - 3| < 5 \end{align*}.
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  8. #8
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by turbozz View Post
    Question: What are the possible values of |2x−3| when 0<|x−1|<2?
    This question and the resulting discussion need a comment on the logic of the question.
    In my view reply #4 is correct.

    Here is the reason. This is a simple implication, an if-then question.
    If 0<|x-1|<2 then what are the possible values of |2 x-3|~?
    If P is true then Q must be true.

    Note that means that if 0<|x-1|<2 then |2x-3|<5.

    If a=1.5 then 0<|a-1|<2 as well as |2a-3|=0<5

    Note that if 0<|x-1|<2 it is possible that |2x-3|<1, example x=1.8.

    It is also worth noting that if x=1 then 0<|x-1|<2 is a false statement.
    BUT that means that if 0<|x-1|<2 then |2x-3|<5 is still a true statement.
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  9. #9
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by Prove It View Post
    The triangle inequality works nicely here to get an upper bound: \displaystyle \begin{align*} |a + b| \leq |a| + |b| \end{align*}

    So \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\leq |2x - 2| + |-1| \\ &= 2|x - 1| + 1 \\ &< 2(2) + 1 \textrm{ since } |x - 1| < 2 \\ &= 5 \end{align*}


    So we can say \displaystyle \begin{align*} |2x - 3| < 5 \end{align*}.

    Also, the reverse triangle inequality is useful here to get a lower bound. \displaystyle \begin{align*} \left| |a| - |b| \right| \leq |a - b| \end{align*}. So that means

    \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\geq \left| |2x - 2| - |1| \right| \textrm{ by the Reverse Triangle Inequality} \\ &= \left| 2|x - 1| - 1 \right| \\ &> \left| 2(0) - 1 \right| \textrm{ since } |x - 1| > 0 \\ &= |-1| \\ &= 1  \end{align*}

    So that means we have \displaystyle \begin{align*} 1 < |2x - 3| < 5 \end{align*}.
    I like the way you showed the solution a lot. Only one issue though shouldn't 0<|2x-3|<5 ?
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  10. #10
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    Re: Find all possible values for the inequality.

    Quote Originally Posted by turbozz View Post
    Only one issue though shouldn't 0<|2x-3|<5 ?
    Did you read reply #8?
    The answer is |2x-3|<5.

    Look at this: 0<|1.5-1|<2 and 0=|2(1.5)-3|<5.
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