Select all the points that are solutions to the following system of inequalities. y ≥ 3x +1 y≥-2+4 Show your work here (0,1) (6,-8) (-2,8) Aa (

To find the solutions to the system of inequalities \(y \geq 3x + 1\) and \(y \geq -2 + 4\), we'll analyze each inequality separately.

1. **First Inequality: \(y \geq 3x + 1\)**  
   - This line has a slope of 3 and a y-intercept of 1. The area above this line (including the line) represents the solutions to this inequality.

2. **Second Inequality: \(y \geq -2 + 4\)**  
   - Simplifying this gives \(y \geq 2\). This represents the area above the horizontal line \(y = 2\).

### Checking Each Point

Let's check the given points one by one to see if they satisfy both inequalities.

1. **Point (0, 1)**  
   - For \(y \geq 3(0) + 1\):  
     \(1 \geq 1\) (True)  
   - For \(y \geq 2\):  
     \(1 \geq 2\) (False)  
   - **Conclusion**: (0, 1) is not a solution.

2. **Point (6, -8)**  
   - For \(y \geq 3(6) + 1\):  
     \(-8 \geq 18 + 1\) (False)  
   - **Conclusion**: (6, -8) is not a solution.

3. **Point (-2, 8)**  
   - For \(y \geq 3(-2) + 1\):  
     \(8 \geq -6 + 1\)  
     \(8 \geq -5\) (True)  
   - For \(y \geq 2\):  
     \(8 \geq 2\) (True)  
   - **Conclusion**: (-2, 8) is a solution.

### Summary of Solutions

The only point that is a solution to the system of inequalities is **(-2, 8)**.

Would you like further details on any part of this process, or do you have questions about the inequalities? Here are some related questions you might consider:

1. How can we graph the inequalities?
2. What are the intersections of the lines defined by the inequalities?
3. How would the solution change if the inequalities were strict (e.g., \(y > 3x + 1\))?
4. What would be the solution set if another inequality were added?
5. Can you explain how to determine if a point is in the solution set graphically?

**Tip:** When graphing inequalities, shade the region that satisfies the inequality and pay attention to whether the boundary line is included (solid line) or not (dashed line).

Select all the points that are solutions to the following system of inequalities: y ≥ 3x + 1 and y ≥ -2 + 4. Check the points (0,1), (6,-8), (-2,8).

Explore how to determine which points are solutions to the system of inequalities y ≥ 3x + 1 and y ≥ 2. This step-by-step guide evaluates the points (0,1), (6,-8), and (-2,8), identifying the valid solutions for students in Grades 9-11.

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