To solve the inequality \( 25 - 4x^2 > 0 \), we can follow these steps:

1. **Rearrange the inequality:**
To make it easier to work with, we can rearrange the inequality:
\[
25 > 4x^2
\]

2. **Divide both sides by 4:**
\[
\frac{25}{4} > x^2
\]
This can also be written as:
\[
x^2 < \frac{25}{4}
\]

3. **Take the square root of both sides:**
Remember when taking the square root of both sides of an inequality, we need to consider both the positive and negative roots:
\[
-\frac{5}{2} < x < \frac{5}{2}
\]

Thus, the solution to the inequality \( 25 - 4x^2 > 0 \) is:
\[
x \in \left(-\frac{5}{2}, \frac{5}{2}\right)
\]

This means that \( x \) can take any value between \(-2.5\) and \(2.5\), not including the endpoints.