Question
Which system of inequalities is represented by the graph?
Responses
⎧⎩⎨⎪⎪y>−12x−2y≤−2x+6y≤x
First line. y is greater than negative fraction 1 half end fraction x minus 2 Second line. y is less than or equal to negative 2 x plus 6 Third line. y is less than or equal to x
⎧⎩⎨⎪⎪y<−2x−2y≤−12x+6y≥x
First line. y is less than negative 2 x minus 2 Second line. y is less than or equal to negative fraction 1 half end fraction x plus 6 Third line. y is greater than or equal to x
⎧⎩⎨⎪⎪y<−12x−2y≤−2x+6y≥x
First line. y is less than negative fraction 1 half end fraction x minus 2 Second line. y is less than or equal to negative 2 x plus 6 Third line. y is greater than or equal to x
⎧⎩⎨⎪⎪y>−2x−2y≤−12x+6y≤x
First line. y is greater than negative 2 x minus 2 Second line. y is less than or equal to negative fraction 1 half end fraction x plus 6 Third line. y is less than or equal to x
A system of 3 linear inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 10 to 10 in increments of 2. The vertical y-axis ranges from negative 10 to 10 in increments of 2. A dashed line y equals negative 2 x minus 2 passes through begin ordered pair negative 1 comma 0 end ordered pair and begin ordered pair 0 comma negative 2 end ordered pair. A solid line y equals negative fraction 1 half end fraction x plus 6 passing through begin ordered pair 0 comma 6 end ordered pair and begin ordered pair 4 comma 4 end ordered pair. A solid line y equals x passes through begin ordered pair 0 comma 0 end ordered pair and begin ordered pair 4 comma 4 end ordered pair. The region above the dashed line y equals negative 2x minus 2, below the solid line y equals x, and below the solid line y equals negative fraction 1 over 2 end fraction x plus 6 is shaded.
Responses
⎧⎩⎨⎪⎪y>−12x−2y≤−2x+6y≤x
First line. y is greater than negative fraction 1 half end fraction x minus 2 Second line. y is less than or equal to negative 2 x plus 6 Third line. y is less than or equal to x
⎧⎩⎨⎪⎪y<−2x−2y≤−12x+6y≥x
First line. y is less than negative 2 x minus 2 Second line. y is less than or equal to negative fraction 1 half end fraction x plus 6 Third line. y is greater than or equal to x
⎧⎩⎨⎪⎪y<−12x−2y≤−2x+6y≥x
First line. y is less than negative fraction 1 half end fraction x minus 2 Second line. y is less than or equal to negative 2 x plus 6 Third line. y is greater than or equal to x
⎧⎩⎨⎪⎪y>−2x−2y≤−12x+6y≤x
First line. y is greater than negative 2 x minus 2 Second line. y is less than or equal to negative fraction 1 half end fraction x plus 6 Third line. y is less than or equal to x
A system of 3 linear inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 10 to 10 in increments of 2. The vertical y-axis ranges from negative 10 to 10 in increments of 2. A dashed line y equals negative 2 x minus 2 passes through begin ordered pair negative 1 comma 0 end ordered pair and begin ordered pair 0 comma negative 2 end ordered pair. A solid line y equals negative fraction 1 half end fraction x plus 6 passing through begin ordered pair 0 comma 6 end ordered pair and begin ordered pair 4 comma 4 end ordered pair. A solid line y equals x passes through begin ordered pair 0 comma 0 end ordered pair and begin ordered pair 4 comma 4 end ordered pair. The region above the dashed line y equals negative 2x minus 2, below the solid line y equals x, and below the solid line y equals negative fraction 1 over 2 end fraction x plus 6 is shaded.
Answers
GPT-4o mini
Based on the description provided, we can analyze the graph to determine the system of inequalities represented.
1. The dashed line \(y = -2x - 2\) indicates that the inequality represented is \(y < -2x - 2\).
2. The solid line \(y = -\frac{1}{2}x + 6\) indicates that the inequality represented is \(y \leq -\frac{1}{2}x + 6\).
3. The solid line \(y = x\) indicates that the inequality represented is \(y \leq x\).
Given these interpretations:
- The first line corresponds to \(y < -2x - 2\).
- The second line corresponds to \(y \leq -\frac{1}{2}x + 6\).
- The third line corresponds to \(y \leq x\).
Therefore, the correct system of inequalities represented by the graph is:
\[
\begin{cases}
y < -2x - 2 \\
y \leq -\frac{1}{2}x + 6 \\
y \leq x
\end{cases}
\]
This matches the third response option:
**\(\begin{cases}
y < -\frac{1}{2}x - 2 \\
y \leq -2x + 6 \\
y \geq x
\end{cases}\)**
So the correct answer is the third response option presented in the question.
1. The dashed line \(y = -2x - 2\) indicates that the inequality represented is \(y < -2x - 2\).
2. The solid line \(y = -\frac{1}{2}x + 6\) indicates that the inequality represented is \(y \leq -\frac{1}{2}x + 6\).
3. The solid line \(y = x\) indicates that the inequality represented is \(y \leq x\).
Given these interpretations:
- The first line corresponds to \(y < -2x - 2\).
- The second line corresponds to \(y \leq -\frac{1}{2}x + 6\).
- The third line corresponds to \(y \leq x\).
Therefore, the correct system of inequalities represented by the graph is:
\[
\begin{cases}
y < -2x - 2 \\
y \leq -\frac{1}{2}x + 6 \\
y \leq x
\end{cases}
\]
This matches the third response option:
**\(\begin{cases}
y < -\frac{1}{2}x - 2 \\
y \leq -2x + 6 \\
y \geq x
\end{cases}\)**
So the correct answer is the third response option presented in the question.