Are the following systems of inequalities the same?Suppose x , y ∈ R, and S 1 is a system of inequalities: S 1 = { x − y ≥ 1 − x + 2 y ≥ 1 3 x − 5 y ≥ 2 } = { x ≥ 1 + y x ≤ 2 y − 1 x ≥ 2 + 5 y 3 } I eliminate x from S 1 to obtain S 2 : S 2 = { 1 + y ≤ 2 y − 1 2 + 5 y 3 ≤ 2 y − 1 } = { 2 ≤ y 2 + 5 y ≤ 6 y − 3 } = { 2 ≤ y 5 ≤ y } Therefore, I know that 5 ≤ y &lt; ∞. Let S 3 denote the following system of inequalities S 3 = { x − y ≥ 1 − x + 2 y ≥ 1 3 x − 5 y ≥ 2 5 ≤ y &lt; ∞ } My question is, is S 3 = S 1 ?

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Are the following systems of inequalities the same?Suppose   x  ,  y  ∈  R, and       S    1   is a system of inequalities:                                    S          1                                    =                  {                                                    x                −                y                ≥                1                                                                    −                x                +                2                y                ≥                1                                                                    3                x                −                5                y                ≥                2                                              }                                                  =                  {                                                    x                ≥                1                +                y                                                                    x                ≤                2                y                −                1                                                                    x                ≥                                                      2                    +                    5                    y                                    3                                                              }                    I eliminate   x from       S    1   to obtain       S    2  :                                    S          2                                    =                  {                                                    1                +                y                ≤                2                y                −                1                                                                                                          2                    +                    5                    y                                    3                                ≤                2                y                −                1                                              }                                                  =                  {                                                    2                ≤                y                                                                    2                +                5                y                ≤                6                y                −                3                                              }                                                  =                  {                                                    2                ≤                y                                                                    5                ≤                y                                              }                    Therefore, I know that   5  ≤  y  &amp;lt;  ∞. Let       S    3   denote the following system of inequalities                                    S          3                                    =                  {                                                    x                −                y                ≥                1                                                                    −                x                +                2                y                ≥                1                                                                    3                x                −                5                y                ≥                2                                                                    5                ≤                y                &amp;lt;                ∞                                              }                    My question is, is       S    3    =      S    1  ?

Donavan Mack · Accepted Answer

Yes,       S    3    =      S    1  . We need only to show that  5  ≤  y  &amp;lt;  ∞Therefore, we have,  3  (  −  x  +  2  y  )  +  3  x  −  5  y  ≥  3  ×  1  +  2    ⟺    y  ≥  5.But, this notation is problematic.  5  ≤  y  ≤  ∞The correct one can be written as follows:  5  ≤  y  &amp;lt;  ∞Or,  y  ∈  [  5  ,  +  ∞  )

Are the following systems of inequalities the same? Suppose x , y ∈ <mi

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