Kyle is thinking of a number greater than -6 2/3 and less than -6 1/2. which number could be kyle's number

The number is greater than - 6  2/3  and less than - 6 1/2.

We focus on the 2/3 and 1/2

Since it is a negative number we think of a fraction less than 2/3 and greater than 1/2, because of the negatives.

2/3 is around 0.66,  1/2 is = 0.5,  Example is 0.6 = 6/10 = 3/5

A number that fits is -6  3/5.

Check:
- 6  2/3 ≈ -6.66
-6  3/5 = - 6.60
-6  1/2  = -6.50

-6.60 is greater than -6.66, and less than -6.50

-6 2/3=-(6*3+2)/3=-20/3=-6.666...≈-6.67
-6 1/2=-(6*2+1)/2=-13/2=-6.5
Therefore:
-6.67<x<-6.5

then:
We have infinites numbers:
x₁=-6.65
x₂=-6.61
x₃=-6.532
x₄=-6.584...

How many shares did he have to begin with?