How can we solve this system of linear inequalities?Let c i be a given non-negative integer for all i ∈ { 1 , … , n }. would like to find the non-negative integers a i and b i for all i ∈ { 1 , … , n } such that: { c i = a i + b i , for all i ∈ { 1 , … , n } a i &lt; a i + 1 , for all i ∈ { 1 , … , n − 1 } b i &gt; b i + 1 , for all i ∈ { 1 , … , n − 1 } b i ≥ 0 and integer , for all i ∈ { 1 , … , n } a i ≥ 0 and integer , for all i ∈ { 1 , … , n }

Question

How can we solve this system of linear inequalities?Let       c    i   be a given non-negative integer for all   i  ∈  {  1  ,  …  ,  n  }.  would like to find the non-negative integers       a    i   and       b    i   for all   i  ∈  {  1  ,  …  ,  n  } such that:                              {                                                                      c                  i                                =                                  a                  i                                +                                  b                  i                                ,                                             for all                 i                ∈                {                1                ,                …                ,                n                }                                                                                      a                  i                                &amp;lt;                                  a                                      i                    +                    1                                                  ,                                             for all                 i                ∈                {                1                ,                …                ,                n                −                1                }                                                                                      b                  i                                &amp;gt;                                  b                                      i                    +                    1                                                  ,                                             for all                 i                ∈                {                1                ,                …                ,                n                −                1                }                                                                                      b                  i                                ≥                0                 and integer                 ,                                             for all                 i                ∈                {                1                ,                …                ,                n                }                                                                                      a                  i                                ≥                0                 and integer                 ,                                             for all                 i                ∈                {                1                ,                …                ,                n                }

Cristian Hamilton · Accepted Answer

What if       c    i    =  0  ∀  i? Then for the first condition you&#039;d need       a    i    =      b    i    =  0  ∀  i. But then the 2nd and 3rd conditions cannot be satisfied.

How can we solve this system of linear inequalities? Let c i </msub> be a given

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Answer & Explanation

New Questions in Algebra I