document.write( "Question 1122569: Let a, b be two positive numbers.
\n" ); document.write( "Decide whether the given statement is true or false. Give a proof or a counterexample.
\n" ); document.write( "(a) If a + b ≤ 1/2, then ((1 − a)/a) * ((1 − b)/b) ≥ 1 .
\n" ); document.write( "(b) If ((1 − a)/a)* ((1 − b)/b) ≥ 1, then a + b ≤ 1/2\r
\n" ); document.write( "\n" ); document.write( "Thank you for your time and help! \n" ); document.write( "
Algebra.Com's Answer #739253 by math_helper(2461)\"\" \"About 
You can put this solution on YOUR website!
a) is true \r
\n" ); document.write( "\n" ); document.write( "a and b positive ==> a > 0, and b > 0
\n" ); document.write( "and \"+a+%2B+b+%3C=+1%2F2+\" means a < 1/2, and b < 1/2
\n" ); document.write( "0 < a < 1/2
\n" ); document.write( "0 < b < 1/2 \r
\n" ); document.write( "\n" ); document.write( "\"+%281-a%29+%3E+1%2F2++\" and \"++a++++%3C+1%2F2+\" \r
\n" ); document.write( "\n" ); document.write( "So \"+%281-a%29%2Fa+%3E+1+\" and similarly \"+%281-b%29%2Fb++%3E+1+\"\r
\n" ); document.write( "\n" ); document.write( "Therefore \"+%28%281-a%29%2Fa%29+%2A+%28%281-b%29%2Fb%29+%3E=+1+\" holds true.\r
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\n" ); document.write( "b) is false. Counterexample: if a=1/4 and b=3/4 then but \"+a%2Bb+=+1%2F4+%2B+3%2F4+=+1++%3E+1%2F2+\"\r
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\n" ); document.write( "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
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