To determine the system of linear inequalities based on the graph description provided, we need to analyze the lines mentioned and their respective shading.
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Dashed Line:
- This line is not included in the solution set, indicating a "greater than" inequality.
- The line passes through the points (2, 1) and (0, -3).
First, we calculate the slope of this line: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{0 - 2} = \frac{-4}{-2} = 2. \] Using the slope-intercept form \(y = mx + b\), starting from point (2, 1): \[ y - 1 = 2(x - 2) \Rightarrow y - 1 = 2x - 4 \Rightarrow y = 2x - 3. \] Therefore, the inequality for the dashed line is: \[ y < 2x - 3. \]
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Solid Line:
- This line is included in the solution set, indicating a "greater than or equal to" inequality.
- The line passes through the points (2, 1) and (-4, 4).
Calculating the slope for this line: \[ \text{slope} = \frac{4 - 1}{-4 - 2} = \frac{3}{-6} = -\frac{1}{2}. \] Using the slope-intercept form starting from point (2, 1): \[ y - 1 = -\frac{1}{2}(x - 2) \Rightarrow y - 1 = -\frac{1}{2}x + 1 \Rightarrow y = -\frac{1}{2}x + 2. \] Therefore, the inequality for the solid line is: \[ y \geq -\frac{1}{2}x + 2. \]
From the shading description, the region is above both the dashed line and the solid line. Thus, we have the following system of linear inequalities:
\[ \begin{aligned}
- & \quad y < 2x - 3 \
- & \quad y \geq -\frac{1}{2}x + 2. \end{aligned} \]
These are the inequalities representing the system depicted in the graph.