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