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System of linear inequalities

  1. Apr 11, 2009 #1
    1. The problem statement, all variables and given/known data

    3431490221_92642e0dd6.jpg

    Determine whether a vector x exists that satisfies the above equation.


    2. Relevant equations
    This is not an actual HW problem, and it is not from a book. (There is no section before or after that explains or introduce this problem or problems similar to it.) So I don't know what topic this problem belongs to. I was wondering what the proper name of this type of problem is called, and what is the normal way (correct way) to solve it.


    3. The attempt at a solution

    At the moment all I could conclude is:

    a > 0
    b > 6a > 0
    c > -11a + 6b > 26a > 0
    d < 0, because:
    -6b-110d +e > 0, since b>0 and e<0 (see below), therefore d must be negative.
    d > 6a - 11b + 110c > 6a - 11b + 110(26a) = 2866a - 11b
    e > 6b + 110d > 36a + 110(2866a - 11b) = 315296a - 1210b

    Since e<0,

    315296a - 1210b < 0
    b > 261a

    An now this screws up the inequality above, where I had "b > 6a".

    I am stumped because I don't know how to combine the equations. Suppose it was not an inequality, so that I am just trying to find a vector such that the product is the zero-vector, then I could conclude that:

    a = 0, therefore:
    b = 0, therefore:
    c = 0, therefore:
    d = 0, therefore:
    e = 0.

    So the only solution for x is trivial. But does this result relate in some way to the inequality?

    I was hoping that there is a proper way to view this problem, or that there is something method that would immediately tell whether the problem has a solution.

    - Thanks
     
  2. jcsd
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