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If $-6\leq a \leq -2$ and $3 \leq b \leq 5$, what is the greatest possible value of $\frac{a + 1/b}{a - 1/b}$?  Express your answer as a common fraction.

 Aug 24, 2023
 #1
avatar+126978 
+1

Note that      [a + 1/b ] / [a -1/b ] =  [  (ab +1) / b ] /  [ (ab  -1) / b ]  =  (ab + 1) / (ab  -1)

 

Note that ab will be negative   and the  numerator and  denominator will always have an  absolute  value difference  of  2

 

The smallest possible  fraction results when   a = -2 and  b = 3  =   -5/- 7 = 5/7

 

The function  is  maximized when a =  -6  and  b  = 5....so....

 

(-6*5 + 1)  / (-6*5  - 1) =      -29   / -31   =  29 / 31

 

 

cool cool cool

 Aug 24, 2023
edited by CPhill  Aug 24, 2023

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