X2-49<0

To solve the inequality \( x^2 - 49 < 0 \), we can start by factoring the left-hand side:

\[ x^2 - 49 = (x - 7)(x + 7) \]

Now, we need to solve the inequality:

\[ (x - 7)(x + 7) < 0 \]

Next, we determine the critical points by setting the factors equal to zero:

\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]

This gives us the intervals to test: \( (-\infty, -7) \), \( (-7, 7) \), and \( (7, \infty) \).

Now, we can test a value from each interval to see where the product \( (x - 7)(x + 7) \) is negative.

1. For the interval \( (-\infty, -7) \), choose \( x = -8 \): \[ (-8 - 7)(-8 + 7) = (-15)(-1) = 15 \quad \text{(positive)} \]

2. For the interval \( (-7, 7) \), choose \( x = 0 \): \[ (0 - 7)(0 + 7) = (-7)(7) = -49 \quad \text{(negative)} \]

3. For the interval \( (7, \infty) \), choose \( x = 8 \): \[ (8 - 7)(8 + 7) = (1)(15) = 15 \quad \text{(positive)} \]

The product \( (x - 7)(x + 7) \) is negative in the interval \( (-7, 7) \).

Thus, the solution to the inequality \( x^2 - 49 < 0 \) is:

\[ \boxed{(-7, 7)} \]