Let's determine the sets [tex]\( S \)[/tex] and [tex]\( T \)[/tex] based on the given conditions and then find their intersection [tex]\( S \cap T \)[/tex].
First, we'll identify the elements in set [tex]\( S \)[/tex]:
[tex]\[ S = \left\{ x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3| - 1}{|x| - 2} \geq 0 \right\} \][/tex]
We need to find the integers within [tex]\([-6, 3]\)[/tex] (excluding [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]), satisfying the inequality:
[tex]\[ \frac{|x+3| - 1}{|x| - 2} \geq 0 \][/tex]
After analyzing this expression, we'll find the set of valid [tex]\(x\)[/tex] values within the working range:
[tex]\[ S = \{ -6, -5, -4, 3 \} \][/tex]
Next, let's identify the elements in set [tex]\( T \)[/tex]:
[tex]\[ T = \left\{ x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0 \right\} \][/tex]
We solve the inequality:
[tex]\[ x^2 - 7|x| + 9 \leq 0 \][/tex]
Solving this, we find:
[tex]\[ T = \{-5, -4, -3, -2, 2, 3, 4, 5\} \][/tex]
Now, we find the intersection [tex]\( S \cap T \)[/tex], which are the common elements between sets [tex]\( S \)[/tex] and [tex]\( T \)[/tex]:
[tex]\[ S \cap T = \{ -5, -4, 3 \} \][/tex]
The number of elements in [tex]\( S \cap T \)[/tex] is:
[tex]\[ |S \cap T| = 3 \][/tex]
Thus, the number of elements in the intersection of sets [tex]\( S \)[/tex] and [tex]\( T \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
