(х-4) (х+2) >=0

To solve the inequality \((x - 4)(x + 2) \geq 0\), we can follow these steps: 1. **Find the critical points**: Set the expression equal to zero to find the critical points. \[ (x - 4)(x + 2) = 0 \] This gives us: - \(x - 4 = 0 \Rightarrow x = 4\) - \(x + 2 = 0 \Rightarrow x = -2\) 2. **Determine the intervals**: The critical points divide the number line into intervals. The intervals to test are: - \((-\infty, -2)\) - \((-2, 4)\) - \((4, \infty)\) 3. **Test each interval**: - For the interval \((-\infty, -2)\), choose a test point like \(x = -3\): \[ (-3 - 4)(-3 + 2) = (-7)(-1) = 7 \geq 0 \quad \text{(true)} \] - For the interval \((-2, 4)\), choose a test point like \(x = 0\): \[ (0 - 4)(0 + 2) = (-4)(2) = -8 \geq 0 \quad \text{(false)} \] - For the interval \((4, \infty)\), choose a test point like \(x = 5\): \[ (5 - 4)(5 + 2) = (1)(7) = 7 \geq 0 \quad \text{(true)} \] 4. **Include the critical points**: Since the inequality is \(\geq 0\), we include the points where the expression equals zero, which are \(x = 4\) and \(x = -2\). 5. **Combine the results**: The solution to the inequality \((x - 4)(x + 2) \geq 0\) is: \[ (-\infty, -2] \cup [4, \infty) \] Thus, the final solution is: \[ x \in (-\infty, -2] \cup [4, \infty) \]