SOLUTION: Let a, b be two positive numbers. Decide whether the given statement is true or false. Give a proof or a counterexample. (a) If a + b ≤ 1/2, then ((1 − a)/a) * ((1 &#

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Let a, b be two positive numbers. Decide whether the given statement is true or false. Give a proof or a counterexample. (a) If a + b ≤ 1/2, then ((1 − a)/a) * ((1 &#      Log On


   

Question 1122569: Let a, b be two positive numbers.
Decide whether the given statement is true or false. Give a proof or a counterexample.
(a) If a + b ≤ 1/2, then ((1 − a)/a) * ((1 − b)/b) ≥ 1 .
(b) If ((1 − a)/a)* ((1 − b)/b) ≥ 1, then a + b ≤ 1/2
Thank you for your time and help!
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
a) is true
a and b positive ==> a > 0, and b > 0
and +a+%2B+b+%3C=+1%2F2+ means a < 1/2, and b < 1/2
0 < a < 1/2
0 < b < 1/2
+%281-a%29+%3E+1%2F2++ and ++a++++%3C+1%2F2+
So +%281-a%29%2Fa+%3E+1+ and similarly +%281-b%29%2Fb++%3E+1+
Therefore +%28%281-a%29%2Fa%29+%2A+%28%281-b%29%2Fb%29+%3E=+1+ holds true.



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+
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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