Question

On a piece of paper, graph this system of inequalities. Then determine which region contains the solution to the system.
y≤-1/3x+3
y≥3x+2

Region A

Region B

Region C

Region D

Answer 1

To graph the system of inequalities given by \( y \leq -\frac{1}{3}x + 3 \) and \( y \geq 3x + 2 \), let’s first identify the boundary lines for each inequality, and then determine the regions that correspond to each inequality.

### Step 1: Graph the Boundary Lines
1. **For the first inequality \( y \leq -\frac{1}{3}x + 3 \)**:
- The boundary line is \( y = -\frac{1}{3}x + 3 \).
- To find points for this line, we can use the y-intercept and the slope.
- When \( x = 0 \): \( y = 3 \) (point \( (0, 3) \))
- When \( x = 3 \): \( y = -\frac{1}{3}(3) + 3 = 2 \) (point \( (3, 2) \))
- Draw a line through these points. Since the inequality includes "less than or equal to," the line should be solid.
- The region below this line represents the solutions to \( y \leq -\frac{1}{3}x + 3 \).

2. **For the second inequality \( y \geq 3x + 2 \)**:
- The boundary line is \( y = 3x + 2 \).
- To find points for this line, again we can use the y-intercept and the slope.
- When \( x = 0 \): \( y = 2 \) (point \( (0, 2) \))
- When \( x = -1 \): \( y = 3(-1) + 2 = -1 \) (point \( (-1, -1) \))
- Draw a line through these points. Since the inequality includes "greater than or equal to," the line should also be solid.
- The region above this line represents the solutions to \( y \geq 3x + 2 \).

### Step 2: Determine the Overlapping Region
- The solution to the system of inequalities is where the shaded regions of both inequalities overlap.
- The region that is below the line \( y = -\frac{1}{3}x + 3 \) and above the line \( y = 3x + 2 \) is where the two inequalities meet.

### Step 3: Identify the Region
To identify which region contains the solution, you can sketch the graph based on the explanations provided above.

Assuming the regions are labeled as:
- **Region A**: Above \( y = -\frac{1}{3}x + 3 \)
- **Region B**: Below \( y = -\frac{1}{3}x + 3 \)
- **Region C**: Below \( y = 3x + 2 \)
- **Region D**: Above \( y = 3x + 2 \)

### Conclusion
The solution will be located in the region where \( y \leq -\frac{1}{3}x + 3 \) (shaded area below that line) and \( y \geq 3x + 2 \) (shaded area above that line) overlap. Without seeing the graph, I would confirm that the overlapping region typically represents where the solutions to those inequalities coexist.

Thus, you should be looking for that intersection or overlapping area to determine the correct region (likely something akin to Region D if it matches the description above).