Relationship between sets defined by linear inequalities












1












$begingroup$


I have the following sets



$$A = {x in [0, 1]^4 mid 83x_1 + 61x_2 + 49x_3 + 20x_4 leq 100}$$



$$B = {x in [0, 1]^4 mid 4x_1 + 3x_2 + 2x_3 + 1x_4 leq 4}$$



How can I prove that $B subset A$?






inequality linear-programming







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$endgroup$








  • 5




    $begingroup$
    Multiply the second inequality by 20.
    $endgroup$
    – Jean Marie
    Jan 28 at 19:41






  • 1




    $begingroup$
    By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
    $endgroup$
    – fleablood
    Jan 28 at 19:50


















1












$begingroup$


I have the following sets



$$A = {x in [0, 1]^4 mid 83x_1 + 61x_2 + 49x_3 + 20x_4 leq 100}$$



$$B = {x in [0, 1]^4 mid 4x_1 + 3x_2 + 2x_3 + 1x_4 leq 4}$$



How can I prove that $B subset A$?






inequality linear-programming







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    Multiply the second inequality by 20.
    $endgroup$
    – Jean Marie
    Jan 28 at 19:41






  • 1




    $begingroup$
    By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
    $endgroup$
    – fleablood
    Jan 28 at 19:50
















1












1








1





$begingroup$


I have the following sets



$$A = {x in [0, 1]^4 mid 83x_1 + 61x_2 + 49x_3 + 20x_4 leq 100}$$



$$B = {x in [0, 1]^4 mid 4x_1 + 3x_2 + 2x_3 + 1x_4 leq 4}$$



How can I prove that $B subset A$?






inequality linear-programming







share|cite|improve this question











$endgroup$




I have the following sets



$$A = {x in [0, 1]^4 mid 83x_1 + 61x_2 + 49x_3 + 20x_4 leq 100}$$



$$B = {x in [0, 1]^4 mid 4x_1 + 3x_2 + 2x_3 + 1x_4 leq 4}$$



How can I prove that $B subset A$?






inequality linear-programming




inequality linear-programming






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 31 at 7:52









Rodrigo de Azevedo

13.2k41960




13.2k41960










asked Jan 28 at 19:38









J. KuhnJ. Kuhn

61




61








  • 5




    $begingroup$
    Multiply the second inequality by 20.
    $endgroup$
    – Jean Marie
    Jan 28 at 19:41






  • 1




    $begingroup$
    By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
    $endgroup$
    – fleablood
    Jan 28 at 19:50
















  • 5




    $begingroup$
    Multiply the second inequality by 20.
    $endgroup$
    – Jean Marie
    Jan 28 at 19:41






  • 1




    $begingroup$
    By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
    $endgroup$
    – fleablood
    Jan 28 at 19:50










5




5




$begingroup$
Multiply the second inequality by 20.
$endgroup$
– Jean Marie
Jan 28 at 19:41




$begingroup$
Multiply the second inequality by 20.
$endgroup$
– Jean Marie
Jan 28 at 19:41




1




1




$begingroup$
By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
$endgroup$
– fleablood
Jan 28 at 19:50






$begingroup$
By proving $4x_1 + 3x_2 + 2x_3 +x_4 le 4 implies 83x_1 + 61x_2 + 49x_3 + 20x_4 le 100$ for $0le x_i le 1$.
$endgroup$
– fleablood
Jan 28 at 19:50












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$begingroup$

Let $x_ige0$. Multiply second inequation by $25$. Then
$$100x_1+75x_2+50x_3+25x_1le100,$$
$$83x_1 + 61x_2 + 49x_3 + 20x_4le 100x_1+75x_2+50x_3+25x_1le100$$






share|cite|improve this answer









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    $begingroup$

    Let $x_ige0$. Multiply second inequation by $25$. Then
    $$100x_1+75x_2+50x_3+25x_1le100,$$
    $$83x_1 + 61x_2 + 49x_3 + 20x_4le 100x_1+75x_2+50x_3+25x_1le100$$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Let $x_ige0$. Multiply second inequation by $25$. Then
      $$100x_1+75x_2+50x_3+25x_1le100,$$
      $$83x_1 + 61x_2 + 49x_3 + 20x_4le 100x_1+75x_2+50x_3+25x_1le100$$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Let $x_ige0$. Multiply second inequation by $25$. Then
        $$100x_1+75x_2+50x_3+25x_1le100,$$
        $$83x_1 + 61x_2 + 49x_3 + 20x_4le 100x_1+75x_2+50x_3+25x_1le100$$






        share|cite|improve this answer









        $endgroup$



        Let $x_ige0$. Multiply second inequation by $25$. Then
        $$100x_1+75x_2+50x_3+25x_1le100,$$
        $$83x_1 + 61x_2 + 49x_3 + 20x_4le 100x_1+75x_2+50x_3+25x_1le100$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 28 at 21:09









        Aleksas DomarkasAleksas Domarkas

        1,62317




        1,62317






























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