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JEE Main & Advanced Mathematics Equations and Inequalities Question Bank Self Evaluation Test - Linear Inequalities

  • question_answer
    A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. The possible length of the shortest board, if the third piece is to be at least 5 cm longer than the second, is

    A) Less than 8 cm

    B) Greater than or equal to 8 cm but less then or equal to 22 cm

    C) Less than 22 cm

    D) Greater than 22 cm  

    Correct Answer: B

    Solution :

    [b] Let the shortest side be x cm.
    Then, by given condition, second length \[=x+3\,\,cm\]
    Third length \[=2xcm\]
    Also given, total length \[=91\]
    Hence, sum of all the three lengths should be less than or equal to 91
    \[\Rightarrow x+x+3+2x\le 91\Rightarrow 4x+3\le 91\]
    Subtracting (-3) to teach tern,
    \[-3+4x+3\le 91-3\]
    \[\Rightarrow 4x\le 88\Rightarrow \frac{4x}{4}\le \frac{88}{4}\Rightarrow x\le \frac{88}{4}\]
    \[\Rightarrow x\le 22cm\]                                      ? (i)
    Again, given that Third length \[\ge \] second length \[+5\]
    \[\Rightarrow 2x\ge (x+3)+5\Rightarrow 2x\ge x+(3+5)\]
    Transferring the term x to L.H.S.,
    \[2x-x\ge 8\Rightarrow x\ge 8\]                              ? (ii)
    From equations (i) and (ii) length of shortest board should be greater than or equal to 8 but less than or equal to 22, i.e., \[8\le x\le 22.\]


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