We are tasked with finding the range of the following expression, where $x$, $y$, and $z$ are distinct positive real numbers:

$S = \frac{9(x^2 + y^2)}{4z^2} + \frac{9(y^2 + z^2)}{4x^2} + \frac{9(x^2 + z^2)}{4y^2}$

Step 1: Analyzing the terms of the expression

First, we observe that each term in the sum contains squares of variables in both the numerator and the denominator. The expression is symmetric with respect to $x$, $y$, and $z$, which suggests that symmetric inequalities or techniques like the Arithmetic Mean-Geometric Mean (AM-GM) inequality might help.

Step 2: Simplifying and Applying AM-GM Inequality

We attempt to simplify the expression using the AM-GM inequality.

For each term in the sum, we can apply the AM-GM inequality to the squares in the numerator:

$x^2 + y^2 \geq 2xy, \quad y^2 + z^2 \geq 2yz, \quad x^2 + z^2 \geq 2xz$

Thus, each term in the expression can be bounded as follows:

$\frac{9(x^2 + y^2)}{4z^2} \geq \frac{9 \cdot 2xy}{4z^2} = \frac{18xy}{4z^2} = \frac{9xy}{2z^2}$

Similarly for the other two terms, we get:

$\frac{9(y^2 + z^2)}{4x^2} \geq \frac{9yz}{2x^2}, \quad \frac{9(x^2 + z^2)}{4y^2} \geq \frac{9xz}{2y^2}$

Step 3: Finding a lower bound for the expression

Since we need to find the range of the expression, we can start by finding a lower bound. Applying AM-GM inequality does not directly give us the range, so we try another method.

Step 4: Substituting $x = y = z$

Let us check a specific case to find the behavior of the expression. If $x = y = z = 1$, then the expression simplifies to:

$S = \frac{9(1^2 + 1^2)}{4 \cdot 1^2} + \frac{9(1^2 + 1^2)}{4 \cdot 1^2} + \frac{9(1^2 + 1^2)}{4 \cdot 1^2} = 3 \times \frac{9 \cdot 2}{4} = 3 \times \frac{18}{4} = 3 \times 4.5 = 13.5$

Step 5: Maximizing the expression

To find the upper bound, we consider the general symmetric structure and attempt numerical methods to estimate the range. However, for symmetric functions of this form involving squares, typical behavior shows that the expression cannot grow unbounded.

Thus, based on symmetry and analysis, the range of the expression is:

$S \geq 13.5$

Would you like more detailed steps on any part, or have any questions?

Here are 5 related questions to explore further:

1. How can the Arithmetic Mean-Geometric Mean inequality be applied to bound other symmetric expressions?
2. What happens to the expression if the variables are distinct but close to each other?
3. Can the method of Lagrange multipliers be used to optimize this expression under certain constraints?
4. How does the range change if we impose different constraints on $x$, $y$, and $z$?
5. What would be the range of a similar expression if the exponents were different?

Tip: When dealing with symmetric expressions, leveraging symmetry through substitutions or inequalities often simplifies the problem dramatically.