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

Question

Prove or disprove that the system of inequalities  a  ,  b  ,  d  ,  e  ,  f  ,  g  ,  h  ,  i  &amp;gt;  0  a  +  e  −  i  &amp;gt;  0  a  e  −  a  i  −  b  d  −  e  i  −  f  h  &amp;gt;  0  −  a  e  i  −  h  f  a  +  b  d  i  −  g  b  f  &amp;gt;  0is inconsistent.

Tanner Hamilton · Accepted Answer

The last two inequalities are inconsistent under the assumption that all the variables are positive (even ignoring   a  +  e  −  i  &amp;gt;  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  &amp;gt;  0  ,which is clearly impossible.