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

    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 $\displaystyle 2x+1$ from all three terms.

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

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

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

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

    Then

    $\displaystyle -1\le\ x<\frac{1}{2}$
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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 $\displaystyle 4x<2x+1\leq 3x+2$, a possible way to solve this is by considering three cases:
    (1)$\displaystyle 4x<2x+1$
    (2)$\displaystyle 4x\leq3x+2$
    (3)$\displaystyle 2x+1\leq3x+2$
    I like this approach.
    Solution sets:
    $\displaystyle \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 $\displaystyle A \cap B \cap C = \left[ { - 1,\frac{1}{2}} \right)$
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  6. #6
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    Re: confused on two inequality sided question

    Hello, GaDoomZ!

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

    Separate the inequalities:

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

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


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


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

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