Determining for what values is a system of inequalities true. <mtable displaystyle="true">

Sonia Ayers 2022-07-03 Answered
Determining for what values is a system of inequalities true.
(1) 8 A 2 α + 44 A α + 60 α > 3 A 2 α 2 + 12 A 2 + 18 A α 2 + 40 A + 27 α 2 + 40
(2) A > 0
(3) α > 1
Is there a procedure that determines for what value range(s) of α inequalities (1) and (2) are satisifed?
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Answers (2)

alomjabpdl0
Answered 2022-07-04 Author has 12 answers
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 ) > 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 > α
So 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 > 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 ) < α 2 ( 3 A 2 + 18 A + 27 ) + ( 12 A 2 + 40 A + 40 )
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Shea Stuart
Answered 2022-07-05 Author has 4 answers
Alternative
8 A 2 α + 44 A α + 60 α > 3 A 2 α 2 + 12 A 2 + 18 A α 2 + 40 A + 27 α 2 + 40
0 > 3 A 2 a 2 > ( 12 A 2 + 18 A α 2 + 40 A + 27 α 2 + 40 ) ( 44 A α + 60 α )
It'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 > 44 A α or in other words let α > 44 18 = 22 9 we have:
( 12 A 2 + 18 A α 2 + 40 A + 27 α 2 + 40 ) ( 44 A α + 60 α ) >
( 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 + 40
Which clearly positive.
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