Question 1122569
a) is true    

a and b positive ==>  a > 0,  and b > 0
and  {{{ a + b <= 1/2 }}} means  a < 1/2,  and b < 1/2
0 < a < 1/2
0 < b < 1/2   

{{{ (1-a) > 1/2  }}}  and   {{{  a    < 1/2 }}} 

So  {{{ (1-a)/a > 1 }}}  and similarly  {{{ (1-b)/b  > 1 }}}

Therefore {{{ ((1-a)/a) * ((1-b)/b) >= 1 }}}  holds true.


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b) is false.   Counterexample:  if a=1/4 and b=3/4 then {{{ ((1-a)/a) *((1-b)/b) = ((3/4)/(1/4))*((1/4)/(3/4)) = (3)*(1/3) = 1  >= 1  }}}  but   {{{ a+b = 1/4 + 3/4 = 1  > 1/2 }}}

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Dear student:   Since (1-a)>1/2 and a<1/2,  (1-a)/a >1.   Think of it this way: X/Y >1 for positive X and Y, when X>Y