System of inequalities with 3 variables a − b + c &gt; 0 a + b + c &lt; 4 9 a − 3 b + c &lt; − 5

Question

System of inequalities with 3 variables                    a        −        b        +        c        &amp;gt;        0                            a        +        b        +        c        &amp;lt;        4                            9        a        −        3        b        +        c        &amp;lt;        −        5

pressacvt · Accepted Answer

Solving the system      {                            a          −          b          +          c          =                      s            1            2                                                a          +          b          +          c          −          4          =          −                      s            2            2                                                9          a          −          3          b          +          c          +          5          =          −                      s            3            2                                  we obtain the solutions      {                            a          =          −                      1            8                    −                      1            8                    (          2                      s            1            2                    +                      s            2            2                    +                      s            3            2                    )                                      b          =          2          −                      1            2                    (                      s            1            2                    +                      s            2            2                    )                                      c          =                      1            8                    (          17          +          6                      s            1            2                    −          3                      s            2            2                    +                      s            3            3                    )                        so we can conclude easily that  a  &amp;lt;  −      1    8    ,        b  &amp;lt;  2

Devin Anderson · Accepted Answer

Since   a  +  c  &amp;gt;  b we get   2  b  &amp;lt;  4 so   b  &amp;lt;  2Also   9  a  −  3  b  +  c  =  (  a  +  c  )  +  8  a  −  3  b  &amp;gt;  8  a  −  2  b so we get   8  a  &amp;lt;  2  b  −  5  &amp;lt;  4  −  5  =  −  1and thus the claim.

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