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Let a and b be positive real numbers such that a + b = 1. Find the set of all possible values of a/b.

 Jun 12, 2024
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First off, we need to put a in terms of b. This helps us identify a range. 

We get

\(a+b=1\\ a=1-b\)

 

Now, plugging that into the expression, we get

\(\frac{1}{a}+\frac{1}{b}= \frac{1}{1-b}+\frac{1}{b}=\frac{b+1-b}{b-b^2}=\frac{1}{b-b^2}\)

 

Now, we can easily find the set of all real numbers. We have

\(\{ R>1\} |\{a,b\} \subset R\) where \(1> a,b > 0\)

 

Thanks! :)

 Jun 12, 2024

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