(x-2)(x+0,5)≥0
To solve the inequality \((x - 2)(x + 0.5) \geq 0\), we can follow these steps:
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 \(x = -0.5\).
Determine the intervals: The critical points divide the number line into three intervals:
- \( (-\infty, -0.5) \)
- \( (-0.5, 2) \)
- \( (2, \infty) \)
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 \( x = 3 \): \[ (3 - 2)(3 + 0.5) = (1)(3.5) = 3.5 \quad \text{(positive)} \]
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: \[ x \in (-\infty, -0.5] \cup [2, \infty) \]
