For what x an y is y ( x 3 + 3 ) 2 &gt; 2 x y - y - 3 ?

Question

For what x an y is             y                        (                      x            3                    +          3          )                2              &amp;gt;          2                        x          y                -        y        -        3            ?

Kaiden Porter · Accepted Answer

Step 1Compacting the inequality we have      -                  y                  (          18          -          x          +          12                      x            3                    +          2                      x            6                    +          3          y          +                      y            2                    )                                                  (            3            +                          x              3                        )                    2                          (          x          -          3          y          -                      y            2                    )                      &amp;gt;    0   or      -                  y                  (          18          -          x          +          12                      x            3                    +          2                      x            6                    +          3          y          +                      y            2                    )                            x        -        3        y        -                  y          2                      &amp;gt;    0  because                     (        3        +                  x          3                )            2        ≥    0    ∀    x    ∈    ℝ  Step 2The feasible set frontier is composed of                    f        1                    (        x        ,        y        )              =    -    y    =    0                      f        2                    (        x        ,        y        )              =    18    -    x    +    12          x      3        +    2          x      6        +    3    y    +          y      2        =    0  and                     f        3                    (        x        ,        y        )              =    x    -    3    y    -          y      2        =    0  The feasible region interior is the set of points             {      x      ,      y      }       obeying                    f        1                    (        x        ,        y        )                            f        2                    (        x        ,        y        )              &amp;gt;    0                and                            f        3                    (        x        ,        y        )              &amp;gt;    0  or                     f        1                    (        x        ,        y        )                            f        2                    (        x        ,        y        )              &amp;lt;    0                and                            f        3                    (        x        ,        y        )              &amp;lt;    0  .

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