How to solve this system of linear inequalities? θ 1 + θ 2 ≥ 0 2 θ 1 + k θ 2 ≥ 0 θ 1 + 3 θ 2 ≤ 0

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How to solve this system of linear inequalities?                              θ          1                +                  θ          2                                    ≥        0                            2                  θ          1                +        k                  θ          2                                    ≥        0                                      θ          1                +        3                  θ          2                                    ≤        0

cefflid6y · Accepted Answer

θ    1    +      θ    2    ≥  0  ⋯  (  i  )and       θ    1    +  3      θ    2    ≤  0    ⟹    −      θ    1    −  3      θ    2    ≥  0  ⋯  (  i  i  )From   (  i  ) and   (  i  i  ),      θ    1    +      θ    2    −      θ    1    −  3      θ    2    ≥  0(  −      θ    2    ≥  0    ⟹        θ    2    ≤  0)Now,      θ    1    +      θ    2    ≥  0    ⟹        θ    1    ≥  −      θ    2    ≥  0(      θ    1    ≥  0)Do similarly with   2      θ    1    +  k      θ    2    ≥  0

ddcon4r · Accepted Answer

Calling              v      →        1    =  (  1  ,  1  )                v      →        2    =  (  2  ,  k  )                v      →        3    =  −  (  1  ,  3  )now if               v      →        2    =      λ    1                v      →        1    +      λ    2                v      →        3   with   (      λ    1    ,      λ    2    )  ≥  0 then the inequality system has non trivial solution otherwise the solution is       θ    1    =      θ    2    =  0. In the present case we have      λ    1    =            6      −      k        2    ≥  0        λ    2    =            2      −      k        2    ≥  0so the condition for non trivial solution is   k  ≤  2Now assuming that the solution is non trivial and calling   Θ  =  (      θ    1    ,      θ    2    ), this solution should verify              v      →        1    ⋅  Θ  ≥  0                v      →        2    ⋅  Θ  ≥  0                v      →        3    ⋅  Θ  ≥  0which is verified for  Θ  (      μ    1    ,      μ    3    )  =      μ    1                v      →        1    +      μ    3                v      →        3    ≥  0with       μ    i    ≥  0  ,  i  =  1  ,  3

How to solve this system of linear inequalities? <mtable columnalign="right left right left righ

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