x-2）（x+7）<0

To solve the inequality \((x - 2)(x + 7) < 0\), we follow these steps:

1. Find the roots: Set the expression equal to zero: \[ (x - 2)(x + 7) = 0 \] The roots are: \[ x - 2 = 0 \Rightarrow x = 2 \] \[ x + 7 = 0 \Rightarrow x = -7 \]

2. Determine intervals: The roots divide the number line into the following intervals:

   • \( (-\infty, -7) \)
   • \( (-7, 2) \)
   • \( (2, \infty) \)

3. Test each interval:

   • For the interval \( (-\infty, -7) \), choose a test point, such as \( x = -8 \): \[ (-8 - 2)(-8 + 7) = (-10)(-1) = 10 \quad (\text{Positive}) \]
   • For the interval \( (-7, 2) \), choose a test point, such as \( x = 0 \): \[ (0 - 2)(0 + 7) = (-2)(7) = -14 \quad (\text{Negative}) \]
   • For the interval \( (2, \infty) \), choose a test point, such as \( x = 3 \): \[ (3 - 2)(3 + 7) = (1)(10) = 10 \quad (\text{Positive}) \]

4. Determine where the product is negative: The product \((x - 2)(x + 7)\) is negative in the interval \( (-7, 2) \).

5. Write the solution: The solution to the inequality \((x - 2)(x + 7) < 0\) is: \[ \boxed{(-7, 2)} \]