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If $-5\leq a \leq -1$ and $1 \leq b \leq 3$, what is the least possible value of $\displaystyle\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) $? Express your answer as a common fraction.

 Feb 17, 2021
 #1
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1/a + 1/b = (b + a) / ab

1/b - 1/a = (a - b) / ab

 

When multiplied, you get: (a + b)(a - b) / a2b2

                                              = (a2 - b2) / a2b2

                                      = 1/b2 - 1/a2


This means you want b = 3, since you want it to be the smallest it can be. Then you want a = -1, since you want it to be the biggest it can be. 

This gives you:
 

1/32 - 1/(-1)2

= 1/9 - 1

= -8/9, which is your answer :)

 Feb 17, 2021

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