x²-25>0

To solve the inequality \( x^2 - 25 > 0 \), we can start by factoring the left-hand side:

\[ x^2 - 25 = (x - 5)(x + 5) \]

Now we have:

\[ (x - 5)(x + 5) > 0 \]

Next, we find the critical points where the expression equals zero:

\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]

The critical points are \( x = -5 \) and \( x = 5 \). These points divide the number line into three intervals:

1. \( (-\infty, -5) \)
2. \( (-5, 5) \)
3. \( (5, \infty) \)

Now we will test a point from each interval to determine where the inequality holds true.

1. Interval \( (-\infty, -5) \): Choose \( x = -6 \): \[ (-6 - 5)(-6 + 5) = (-11)(-1) = 11 > 0 \] So, the inequality holds in this interval.

2. Interval \( (-5, 5) \): Choose \( x = 0 \): \[ (0 - 5)(0 + 5) = (-5)(5) = -25 < 0 \] So, the inequality does not hold in this interval.

3. Interval \( (5, \infty) \): Choose \( x = 6 \): \[ (6 - 5)(6 + 5) = (1)(11) = 11 > 0 \] So, the inequality holds in this interval.

The solution to the inequality \( x^2 - 25 > 0 \) is:

\[ x \in (-\infty, -5) \cup (5, \infty) \]

In interval notation, this solution can be expressed as:

\[ (-\infty, -5) \cup (5, \infty) \]