Question
What system of linear inequalities is shown in the graph?
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A system of 2 linear inequalities graphed on a coordinate plane. The horizontal x axis ranges from negative 5 to 5 in increments of 1. The vertical y axis ranges from negative 5 to 5 in increments of 1. A dashed line passes through begin ordered pair 2 comma 1 end ordered pair and begin ordered pair 0 comma negative 3 end ordered pair. A solid line passes through begin ordered pair 2 comma 1 end ordered pair and begin ordered pair negative 4 comma 4 end ordered pair. The region above the dashed line and above the solid line is shaded.
Enter your answers in the boxes.
$\ngtr$$\nless$
A system of 2 linear inequalities graphed on a coordinate plane. The horizontal x axis ranges from negative 5 to 5 in increments of 1. The vertical y axis ranges from negative 5 to 5 in increments of 1. A dashed line passes through begin ordered pair 2 comma 1 end ordered pair and begin ordered pair 0 comma negative 3 end ordered pair. A solid line passes through begin ordered pair 2 comma 1 end ordered pair and begin ordered pair negative 4 comma 4 end ordered pair. The region above the dashed line and above the solid line is shaded.
Answers
GPT-4o mini
To determine the system of linear inequalities based on the graph description provided, we need to analyze the lines mentioned and their respective shading.
1. **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:
\[
y < 2x - 3.
\]
2. **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:
\[
\begin{aligned}
1. & \quad y < 2x - 3 \\
2. & \quad y \geq -\frac{1}{2}x + 2.
\end{aligned}
\]
These are the inequalities representing the system depicted in the graph.
1. **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:
\[
y < 2x - 3.
\]
2. **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:
\[
\begin{aligned}
1. & \quad y < 2x - 3 \\
2. & \quad y \geq -\frac{1}{2}x + 2.
\end{aligned}
\]
These are the inequalities representing the system depicted in the graph.