(x-2)(x+0,5)≥0

To solve the inequality \((x - 2)(x + 0.5) \geq 0\), we can follow these steps: 1. **Find the critical points**: Set each factor equal to zero: \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x + 0.5 = 0 \quad \Rightarrow \quad x = -0.5 \] The critical points are \(x = 2\) and \(x = -0.5\). 2. **Determine the intervals**: The critical points divide the number line into three intervals: - \( (-\infty, -0.5) \) - \( (-0.5, 2) \) - \( (2, \infty) \) 3. **Test each interval**: Choose a test point from each interval to determine the sign of the product in that interval. - For the interval \( (-\infty, -0.5) \), choose \( x = -1 \): \[ (-1 - 2)(-1 + 0.5) = (-3)(-0.5) = 1 \quad \text{(positive)} \] - For the interval \( (-0.5, 2) \), choose \( x = 0 \): \[ (0 - 2)(0 + 0.5) = (-2)(0.5) = -1 \quad \text{(negative)} \] - For the interval \( (2, \infty) \), choose \( x = 3 \): \[ (3 - 2)(3 + 0.5) = (1)(3.5) = 3.5 \quad \text{(positive)} \] 4. **Conclusion**: The product \((x - 2)(x + 0.5)\) is: - Positive on the intervals \( (-\infty, -0.5) \) and \( (2, \infty) \) - Negative on the interval \( (-0.5, 2) \) Since the inequality is \((x - 2)(x + 0.5) \geq 0\), we include the points where the expression is zero: - At \(x = -0.5\) and \(x = 2\), the expression is zero. Thus, the solution to the inequality \((x - 2)(x + 0.5) \geq 0\) is: \[ x \in (-\infty, -0.5] \cup [2, \infty) \]