Difference of Squares

To get the minimum value, you take the smallest value for each variable.  So the minimum value is (1/(-5)) + 1/1)(1/(-5) - 1/1) = 24/25.

$\displaystyle\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ \displaystyle\left(\frac{1}{b}+\frac{1}{a}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ = \displaystyle\left(\frac{1}{b}\right)^2-\left(\frac{1}{a}\right)^2\\$

For the smallest value you want the absolute value of  b to be as big as possible,  and the absolute value of a to be as small as possible but neither can be 0

b=3,  a= -1

1/9  -  1 = -8/9