To determine a true statement about the graphed system of inequalities \( y > -\frac{1}{3}x \) and \( y < -\frac{1}{3}x - 2 \), we need to analyze these inequalities.
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First Inequality: \( y > -\frac{1}{3}x \)
- This represents the region above the line \( y = -\frac{1}{3}x \). The line itself is not included in the solution (since it's a "greater than" inequality), so it is dashed.
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Second Inequality: \( y < -\frac{1}{3}x - 2 \)
- This represents the region below the line \( y = -\frac{1}{3}x - 2 \). Again, this line is dashed because it is a "less than" inequality.
Key Observations
- The slopes of both lines are the same, which is \( -\frac{1}{3} \). This means they are parallel.
- The second line \( y = -\frac{1}{3}x - 2 \) is situated below the first line \( y = -\frac{1}{3}x \) because the y-intercept of the second line is lower (specifically at -2, compared to 0 for the first line).
Conclusion
Since the region described by the first inequality (above the first line) does not intersect with the region described by the second inequality (below the second line), there are no solutions that satisfy both inequalities simultaneously.
True Statement
- There are no points that satisfy both inequalities simultaneously; the regions defined by the inequalities do not overlap.
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