(х-3)(х+5)>0

To solve the inequality \((x - 3)(x + 5) > 0\), we can follow these steps:

1. Identify critical points: Set the expression to zero to find critical values.

   \[ (x - 3)(x + 5) = 0 \]

   This gives us two critical points:

   • \(x - 3 = 0 \quad \Rightarrow \quad x = 3\)
   • \(x + 5 = 0 \quad \Rightarrow \quad x = -5\)

2. Divide the number line: The critical points divide the number line into intervals. The intervals are:

   • \( (-\infty, -5) \)
   • \( (-5, 3) \)
   • \( (3, \infty) \)

3. Test each interval: Choose a test point from each interval to determine where the product is positive.

   • Interval \( (-\infty, -5) \): Choose \(x = -6\) \[ (-6 - 3)(-6 + 5) = (-9)(-1) = 9 > 0 \]
   • Interval \( (-5, 3) \): Choose \(x = 0\) \[ (0 - 3)(0 + 5) = (-3)(5) = -15 < 0 \]
   • Interval \( (3, \infty) \): Choose \(x = 4\) \[ (4 - 3)(4 + 5) = (1)(9) = 9 > 0 \]

4. Write the solution: The product \((x - 3)(x + 5) > 0\) is satisfied in the intervals where we found positive values:

   • \( (-\infty, -5) \)
   • \( (3, \infty) \)

5. Conclusion: Thus, the solution to the inequality \((x - 3)(x + 5) > 0\) is: \[ x \in (-\infty, -5) \cup (3, \infty) \]