Animated Solution for Mathematics - Basic Mathematics: Let $S=\{x\in [-6,3]-\{-2,2\}:\frac{|x+3|-1}{|x|-2}\ge 0\}$ and $T=\{x\in \mathbb{Z}:x^2-7|x|+9\le 0\}$. Then the number of elements in $S\cap T$ is

Welcome, fellow traveler of the mathematical landscape! Today, we are embarking on a journey to solve a problem that tests your precision, logic, and ability to navigate the traps of absolute values and rational inequalities.

We are tasked with finding the intersection $S\cap T$ for two sets defined under specific constraints. Let us begin our systematic approach.

The Rational Inequality (Set $S$)

Our first challenge is to define Set $S$ using the condition:

The domain is constrained to $x\in [-6,3]\setminus \{-2,2\}$.

To solve this, we identify the critical points. The numerator $|x+3|-1$ vanishes when $|x+3|=1$, which yields $x=-2$ or $x=-4$.

The denominator $|x|-2$ vanishes when $|x|=2$, giving $x=2$ or $x=-2$. Since the denominator cannot be zero, $x=2$ and $x=-2$ are strictly excluded from the solution set.

Using the Wavy Curve Method across the critical points $\{-4,-2,2\}$, we determine the expression is non-negative when $x\in (-\infty ,-4]\cup (2,\infty )$.

Intersecting this result with the domain $x\in [-6,3]\setminus \{-2,2\}$, we obtain:

The Quadratic Elegance (Set $T$)

Now, we define Set $T$ by the condition $x^2-7|x|+9\le 0$, where $x\in \mathbb{Z}$. Noting that $x^2=|x{|}^2$, we substitute $t=|x|$ to get the quadratic inequality:

Applying the quadratic formula, the roots are:

Given $\sqrt{13}\approx 3.6$, the roots are approximately $1.7$ and $5.3$. Thus, the inequality holds for $1.7\le |x|\le 5.3$.

Since $x$ must be an integer, $|x|$ can only take the values $2,3,4,$ or $5$. Consequently, the set $T$ is:

The Final Convergence

We now find the intersection $S\cap T$ by checking which elements of $T$ reside within the intervals of $S=[-6,-4]\cup (2,3]$.

Evaluating the elements of $T$: $-5$ is in $[-6,-4]$. $-4$ is in $[-6,-4]$. $3$ is in $(2,3]$. The values $-3,-2,2,4,$ and $5$ do not satisfy the conditions of $S$.

Thus, the intersection is:

The number of elements in the intersection is 3. We have successfully conquered the problem through rigorous logical steps.