To determine the graph that represents the solution set of the inequality \( y > -\frac{3}{2}x - 1 \), we can start by analyzing the equation of the line associated with the inequality.
Steps to Analyze the Inequality:
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Identify the Slope and Y-intercept: The inequality is in the form \( y > mx + b \), where \( m = -\frac{3}{2} \) and \( b = -1 \).
- The slope is \(-\frac{3}{2}\), which means that for every 2 units you move right (along the x-axis), you move down 3 units (along the y-axis).
- The y-intercept is \(-1\), which means the line crosses the y-axis at the point \((0, -1)\).
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Graph the Boundary Line:
- First, you would graph the line \( y = -\frac{3}{2}x - 1 \).
- Since the inequality is "greater than" (>), and not "greater than or equal to" (≥), the line will be a dashed line indicating that points on the line are not included in the solution.
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Shade the Appropriate Area:
- To find which area should be shaded, select a test point that is not on the line (for example, the point \((0, 0)\)).
- Substitute this point into the inequality: \[ 0 > -\frac{3}{2}(0) - 1 \implies 0 > -1 \]
- This is a true statement, so we shade the area that includes the point \((0, 0)\).
Conclusion:
- The solution set is the area above the dashed line \( y = -\frac{3}{2}x - 1 \), which corresponds to values of \( y \) that are greater than the line.
When you look at the graphs provided, the one that features a dashed line for \( y = -\frac{3}{2}x - 1 \) with shading above the line is the correct representation of the solution set of the given inequality \( y > -\frac{3}{2}x - 1 \).
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