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Math Help - Rewrite as a double inequality?

  1. #1
    Super Member sakonpure6's Avatar
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    Rewrite as a double inequality?

    I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".

    My attempt:

    5x+7 < 10 and (-5x+7)<10
    5x < 3 and -5x<17
    x <3/5 and x>17/ 5

    Is that the solution? The given options are :

    a. -10> 5x +7 >10
    b. -10< 5x+7<10
    c. -10 < 5x+7>10
    d. 0<5x+7<10

    (I know c and a are wrong because of the direction of the greater than sign. To me b. seems right)
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  2. #2
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    Re: Rewrite as a double inequality?

    Quote Originally Posted by sakonpure6 View Post
    I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".
    This depends upon this theorem: |a|\le |b|\text{ if and only if }-|b|\le a\le |b| .

    So in this case -10<5x+7<10.
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  3. #3
    Super Member sakonpure6's Avatar
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    Re: Rewrite as a double inequality?

    So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem. Thank you!
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    Re: Rewrite as a double inequality?

    Quote Originally Posted by sakonpure6 View Post
    So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem.
    It is such a basic property of the metric that it has no name.
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