What is the solution set for the inequality? \( (-\infty, 2] \) \( \left(-\infty, \frac{34}{9}\right] \) \( [2, \infty) \) \( \left[\frac{34}{9}, \infty\right) \)

To find the solution set for the inequality, we need to analyze the given intervals: 1. **Interval \( (-\infty, 2] \)**: This includes all numbers less than or equal to 2. 2. **Interval \( \left(-\infty, \frac{34}{9}\right] \)**: This includes all numbers less than or equal to \(\frac{34}{9}\). 3. **Interval \( [2, \infty) \)**: This includes all numbers greater than or equal to 2. 4. **Interval \( \left[\frac{34}{9}, \infty\right) \)**: This includes all numbers greater than or equal to \(\frac{34}{9}\). Next, we need to determine the relationship between these intervals, particularly focusing on the critical points \(2\) and \(\frac{34}{9}\). ### Step 1: Compare the critical points First, we calculate \(\frac{34}{9}\): \[ \frac{34}{9} \approx 3.78 \] ### Step 2: Analyze the intervals - The interval \( (-\infty, 2] \) includes all values up to 2. - The interval \( \left(-\infty, \frac{34}{9}\right] \) includes all values up to approximately 3.78. - The interval \( [2, \infty) \) includes all values starting from 2. - The interval \( \left[\frac{34}{9}, \infty\right) \) includes all values starting from approximately 3.78. ### Step 3: Determine the union of intervals Now, we can combine these intervals based on their overlaps: - The first two intervals \( (-\infty, 2] \) and \( \left(-\infty, \frac{34}{9}\right] \) can be combined into \( (-\infty, \frac{34}{9}] \). - The last two intervals \( [2, \infty) \) and \( \left[\frac{34}{9}, \infty\right) \) can be combined into \( [2, \infty) \). ### Final Solution Set Thus, the overall solution set for the inequality is: \[ (-\infty, \frac{34}{9}] \cup [2, \infty) \] This means the solution set includes all values less than or equal to \(\frac{34}{9}\) and all values greater than or equal to 2. If you need further assistance or a specific inequality to solve, please provide that information!