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Math Help - confused on two inequality sided question

  1. #1
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    confused on two inequality sided question

    4x<2x+1<//<3x+2

    I simplify it down to 2x<1<//<1x+2

    but I don't know where to go from there
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  2. #2
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    Re: confused on two inequality sided question

    the second symbol is a less than or equal to sign I am posting from a tablet so didn't have the option
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: confused on two inequality sided question

    The intequality is 4x<2x+1\leq 3x+2, a possible way to solve this is by considering three cases:
    (1) 4x<2x+1
    (2) 4x\leq3x+2
    (3) 2x+1\leq3x+2

    By solving this three inequality's and summarizing the solutions you will get the solutions for the original inequality.
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  4. #4
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    Re: confused on two inequality sided question

    Quote Originally Posted by GaDoomZ View Post
    4x<2x+1<//<3x+2

    I simplify it down to 2x<1<//<1x+2

    but I don't know where to go from there
    You could subtract 2x+1 from all three terms.

    4x-(2x+1)<2x+1-(2x+1)\le\ 3x+2-(2x+1)

    2x-1<0\le\ x+1

    2x-1<0 is saying 2x<1\Rightarrow\ x<\frac{1}{2}

    0\le\ x+1 is saying x\ge\ -1

    Then

    -1\le\ x<\frac{1}{2}
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  5. #5
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    Re: confused on two inequality sided question

    Quote Originally Posted by Siron View Post
    The intequality is 4x<2x+1\leq 3x+2, a possible way to solve this is by considering three cases:
    (1) 4x<2x+1
    (2) 4x\leq3x+2
    (3) 2x+1\leq3x+2
    I like this approach.
    Solution sets:
    \begin{gathered}  (1)\;A = \left( { - \infty ,\frac{1}{2}} \right) \hfill \\  (2)\;B = \left( { - \infty ,2} \right] \hfill \\  (3)\;C = \left[ { - 1,\infty } \right) \hfill \\ \end{gathered}

    So A \cap B \cap C = \left[ { - 1,\frac{1}{2}} \right)
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  6. #6
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    Re: confused on two inequality sided question

    Hello, GaDoomZ!

    4x \:<\:2x+1 \:\le\: 3x+2

    Separate the inequalities:

    . . 4x \:<\:2x+1 \quad\Rightarrow\quad 2x \:<\:1 \quad\Rightarrow\quad x \:<\:\tfrac{1}{2}

    . . 2x+1\:\le\:3x+2 \quad\Rightarrow\quad -x \:\le\:1 \quad\Rightarrow\quad x \:\ge\:-1


    Therefore: . -1\:\le\:x\:<\:\tfrac{1}{2}


    . . \begin{array}{cccccc}---\!\!\!\!\! & \bullet\!\!\!\!\! & ===\!\!\!\!\! & \circ\!\!\!\!\! & ---\!\!\!\!\! \\ & \;\:\text{-}1 && \quad\frac{1}{2} \end{array}

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