Given a system of inequalities A x ≤ b , how can I derive the upper and lower bound on x?I have the following system of inequalities A x ≤ b, where x 1 and x 2 are unknown, and the a's and и's are constants. [ a 11 a 12 a 21 a 22 ] [ x 1 x 2 ] ≤ [ b 1 b 2 ]

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Given a system of inequalities    A  x  ≤  b , how can I derive the upper and lower bound on    x?I have the following system of inequalities   A  x  ≤  b, where       x    1   and       x    2   are unknown, and the   a&#039;s and   и&#039;s are constants.      [                                        a                      11                                                a                      12                                                            a                      21                                                a                      22                                ]        [                                        x            1                                                            x            2                                ]    ≤      [                                        b            1                                                            b            2                                ]

floygdarvn · Accepted Answer

Given                              a                  11                          x          1                +                  a                  12                          x          2                                    ≤                  b          1                                              a                  21                          x          1                +                  a                  22                          x          2                                    ≤                  b          2                    Suppose       a      11   and       a      21    &amp;gt;  0  ,, then                              x          1                                    ≤                                            b              1                        −                          a                          12                                      x              2                                            a                      11                                                        x          1                                    ≤                                            b              2                        −                          a                          22                                      x              2                                            a                      21                              Suppose       a      11   and       a      21    &amp;lt;  0  ,, then                              x          1                                    ≥                                            a                          12                                      x              2                        −                          b              1                                            a                      11                                                        x          1                                    ≥                                            a                          22                                      x              2                        −                          b              2                                            a                      21                              The upper bound of       x    1   is   m  i  n      (                            b          1                −                  a                  12                          x          2                            a              11              ,                            b          2                −                  a                  22                          x          2                            a              21              )  , and the lower bound of       x    1   is   m  a  x      (                            a                  12                          x          2                −                  b          1                            a              11              ,                            a                  22                          x          2                −                  b          2                            a              21              )

reryfaikear · Accepted Answer

You are looking for Fourier–Motzkin elimination, of which you are performing just the first step. The next step is concluding that if the lower bound on       x    1   is not greater than the upper bound on       x    1  , you know that a feasible       x    1   exists, and you no longer need to specify its value, leaving you with inequalities for just       x    2  .

Given a system of inequalities A x ≤ b , how can I derive the upper and

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