\( \frac { y ^ { 3 } + 7 } { y ^ { 3 } - 7 } + \frac { y ^ { 3 } - 7 } { r ^ { 3 } + 7 } \geq \frac { 49 ^ { 2 } + 96 } { 49 ^ { 2 } - 49 } \)

To tackle this kind of inequality, let's first investigate the fundamental nature of the expressions involved. The left side contains fractions where each term has a similar structure, which hints at the possibility of combining them into a single fraction for simplification. Moreover, when evaluating \(49^2\) and \(96\), we can calculate \(49^2 = 2401\), so the right side becomes \( \frac{2401 + 96}{2401 - 49} = \frac{2497}{2352} \). Now, the crux of the solution lies in understanding how changing values of \(y\) (and possibly \(r\) if implicitly introduced) can affect the overall inequality's validity. If we delve deeper into the terms \(y^3\) and \(r^3\), we'll find that they share common properties that can help construct an effective strategy for handling this inequality. Consider substituting some values for \(y\) or transforming the expressions into a simpler form to see if the inequality holds under various scenarios, facilitating a conceptual leap toward finding conditions under which the original inequality is satisfied. This will lead us to not only solve but understand the intrinsic relationships among the variables. Additionally, don't hesitate to test boundary conditions or utilize graphical methods to visualize how \(y\) and \(r\) interact with the given expressions! It can transform complex calculations into tangible insights.