Determining for what values is a system of inequalities true. (1) 8 A 2 α + 44 A α + 60 α &gt; 3 A 2 α 2 + 12 A 2 + 18 A α 2 + 40 A + 27 α 2 + 40 (2) A &gt; 0 (3) α &gt; 1 Is there a procedure that determines for what value range(s) of α inequalities (1) and (2) are satisifed?

Question

Determining for what values is a system of inequalities true.                    (1)                    8                  A          2                α        +        44        A        α        +        60        α        &amp;gt;        3                  A          2                          α          2                +        12                  A          2                +        18        A                  α          2                +        40        A        +        27                  α          2                +        40                                (2)                    A        &amp;gt;        0                                (3)                    α        &amp;gt;        1            Is there a procedure that determines for what value range(s) of   α inequalities (1) and (2) are satisifed?

alomjabpdl0 · Accepted Answer

LHS is   α  (  8      A    2    +  44  A  +  60  ) and RHS is       α    2    (  3      A    2    +  18  A  +  27  )  +  (  12      A    2    +  40  A  +  40  ). And as all components are positive:So   α  (  8      A    2    +  44  A  +  60  )  &amp;gt;      α    2    (  3      A    2    +  18  A  +  27  )  +  (  12      A    2    +  40  A  +  40  )    ⟺              8              A        2            +      44      A      +      60              3              A        2            +      18      A      +      27        −      1    α              12              A        2            +      40      A      +      40              3              A        2            +      18      A      +      27        &amp;gt;  αSo this will be false whenever   α  ≥            8              A        2            +      44      A      +      60              3              A        2            +      18      A      +      27        −      1    α              12              A        2            +      40      A      +      40              3              A        2            +      18      A      +      27       which will include (but not be restricted to) whenever   α  ≥            8              A        2            +      44      A      +      60              3              A        2            +      18      A      +      27      .For any   A  &amp;gt;  0 we can always find   α  ≥            8              A        2            +      44      A      +      60              3              A        2            +      18      A      +      27       is which case  α  (  8      A    2    +  44  A  +  60  )  &amp;lt;      α    2    (  3      A    2    +  18  A  +  27  )  +  (  12      A    2    +  40  A  +  40  )

Shea Stuart · Accepted Answer

Alternative  8      A    2    α  +  44  A  α  +  60  α  &amp;gt;  3      A    2        α    2    +  12      A    2    +  18  A      α    2    +  40  A  +  27      α    2    +  40    ⟺    0  &amp;gt;  −  3      A    2        a    2    &amp;gt;  (  12      A    2    +  18  A      α    2    +  40  A  +  27      α    2    +  40  )  −  (  44  A  α  +  60  α  )It&#039;s pretty clear that if we take   A and   α large enough we can find values where the RHS is positive.For example: If we let   18  A      α    2    &amp;gt;  44  A  α or in other words let   α  &amp;gt;      44    18    =      22    9   we have:  (  12      A    2    +  18  A      α    2    +  40  A  +  27      α    2    +  40  )  −  (  44  A  α  +  60  α  )  &amp;gt;  (  12      A    2    +  44  A  α  +  40  A  +  27  ⋅      22    9    α  +  40  )  −  (  44  A  α  +  60  α  )  =  12      A    2    +  40  A  +  66  α  +  40  )  −  60  α  =  12      A    2    +  40  A  +  6  α  +  40Which clearly positive.

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