Kyle Sutton

2022-07-14

Prove or disprove that the system of inequalities
$a,b,d,e,f,g,h,i>0$
$a+e-i>0$
$ae-ai-bd-ei-fh>0$
$-aei-hfa+bdi-gbf>0$
is inconsistent.

Tanner Hamilton

Expert

The last two inequalities are inconsistent under the assumption that all the variables are positive (even ignoring $a+e-i>0$). If the last two inequalities were both true, then the last inequality plus $i$ times the second-to-last inequality would also be true; but this equals
$\left(-aei-hfa+bdi-gbf\right)+i\left(ae-ai-bd-ei-fh\right)=-hfa-gbf-a{i}^{2}-e{i}^{2}-fhi>0,$
which is clearly impossible.

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