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
y≤-1/3x+3
y≥3x+2
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GPT-4o mini
To graph the system of inequalities \( y \leq -\frac{1}{3}x + 3 \) and \( y \geq 3x + 2 \), follow these steps:
### Step 1: Graph the Boundary Lines
1. **For the first inequality \( y \leq -\frac{1}{3}x + 3 \)**:
- This is a linear equation with a slope of \(-\frac{1}{3}\) and a y-intercept of \(3\).
- To graph the line, plot the y-intercept at \( (0, 3) \).
- Use the slope to find another point: from \( (0, 3) \), move down 1 unit and right 3 units to reach \( (3, 2) \).
- Draw a dashed line (since the inequality is \(\leq\), which does not include equality).
- Shade the region below the line since the inequality is \( y \leq \).
2. **For the second inequality \( y \geq 3x + 2 \)**:
- This is a linear equation with a slope of \(3\) and a y-intercept of \(2\).
- Plot the y-intercept at \( (0, 2) \).
- Use the slope to find another point: from \( (0, 2) \), move up 3 units and right 1 unit to reach \( (1, 5) \).
- Draw a solid line (since the inequality is \(\geq\), which includes equality).
- Shade the region above the line since the inequality is \( y \geq \).
### Step 2: Identify the Solution Region
- The solution to the system of inequalities is where the shaded areas overlap. To find this region, look for the points that satisfy both inequalities simultaneously.
- By examining the graphs, observe where the area shaded for \( y \leq -\frac{1}{3}x + 3 \) and the area shaded for \( y \geq 3x + 2 \) intersect.
### Conclusion
- The final solution region will be where both conditions are met (the intersection of both shaded areas).
- If there’s no overlapping region (depending on the slopes and y-intercepts), then there could be no solution to the system.
### Note:
To complete the task, you would typically draw this on a graph to visualize the solution. However, the nature of both inequalities indicates that it is likely no overlapping region exists because one line slopes downward and the other slopes upward and typically, given their slopes and intercepts, they do not intersect within the first coordinate quadrants where we commonly analyze linear inequalities in this context. This needs to be confirmed by sketching the graphs for a precise interpretation.
### Step 1: Graph the Boundary Lines
1. **For the first inequality \( y \leq -\frac{1}{3}x + 3 \)**:
- This is a linear equation with a slope of \(-\frac{1}{3}\) and a y-intercept of \(3\).
- To graph the line, plot the y-intercept at \( (0, 3) \).
- Use the slope to find another point: from \( (0, 3) \), move down 1 unit and right 3 units to reach \( (3, 2) \).
- Draw a dashed line (since the inequality is \(\leq\), which does not include equality).
- Shade the region below the line since the inequality is \( y \leq \).
2. **For the second inequality \( y \geq 3x + 2 \)**:
- This is a linear equation with a slope of \(3\) and a y-intercept of \(2\).
- Plot the y-intercept at \( (0, 2) \).
- Use the slope to find another point: from \( (0, 2) \), move up 3 units and right 1 unit to reach \( (1, 5) \).
- Draw a solid line (since the inequality is \(\geq\), which includes equality).
- Shade the region above the line since the inequality is \( y \geq \).
### Step 2: Identify the Solution Region
- The solution to the system of inequalities is where the shaded areas overlap. To find this region, look for the points that satisfy both inequalities simultaneously.
- By examining the graphs, observe where the area shaded for \( y \leq -\frac{1}{3}x + 3 \) and the area shaded for \( y \geq 3x + 2 \) intersect.
### Conclusion
- The final solution region will be where both conditions are met (the intersection of both shaded areas).
- If there’s no overlapping region (depending on the slopes and y-intercepts), then there could be no solution to the system.
### Note:
To complete the task, you would typically draw this on a graph to visualize the solution. However, the nature of both inequalities indicates that it is likely no overlapping region exists because one line slopes downward and the other slopes upward and typically, given their slopes and intercepts, they do not intersect within the first coordinate quadrants where we commonly analyze linear inequalities in this context. This needs to be confirmed by sketching the graphs for a precise interpretation.
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