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Prove or disprove this statement

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Consider the statement that if a and b are real numbers in the open interval (0, 1), then a/[b(1 - a)] > 1.

[Note: To say that a and b are in the open interval (0, 1) means that 0 < a < 1 and 0 < b < 1.]

Either prove this statement (that is, prove that it's true for all real numbers a, b in the open interval (0, 1)) or give a counterexample (that is, give specific real numbers a, b such that a/[b(1 - a)] <= 1).

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Solution Summary

A detailed explanation of this (either a proof of the given statement if it is true, or a counterexample if it is false) is provided.

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Let's look at an individual case.

Let's try some value for a (for example, a = 1/6). Then a/(1 - a) = (1/6)/[1 - (1/6)] = (1/6)/(5/6) = (1/6)(6/5) = 1/5.

For this value of a (that is, a = 1/6), can we come up with some number b in the open interval (0, 1) that will make the given statement FALSE? ...

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  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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