Prove or disprove that the system of inequalities a , b , d , e...

Kyle Sutton

Kyle Sutton

Answered

2022-07-14

Prove or disprove that the system of inequalities
a , b , d , e , f , g , h , i > 0
a + e i > 0
a e a i b d e i f h > 0
a e i h f a + b d i g b f > 0
is inconsistent.

Answer & Explanation

Tanner Hamilton

Tanner Hamilton

Expert

2022-07-15Added 12 answers

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
( a e i h f a + b d i g b f ) + i ( a e a i b d e i f h ) = h f a g b f a i 2 e i 2 f h i > 0 ,
which is clearly impossible.

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