(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: \[ x \in (-\infty, -5) \cup (3, \infty) \]