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 &#
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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) (Show Source): You can put this solution on YOUR website!
a) is true
a and b positive ==> a > 0, and b > 0
and means a < 1/2, and b < 1/2
0 < a < 1/2
0 < b < 1/2
and
So and similarly
Therefore holds true.
b) is false. Counterexample: if a=1/4 and b=3/4 then but
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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
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