Question
Which system of inequalities represents the graph?
Responses
⎧⎩⎨⎪⎪5y−10≥−3xy+5>xy+5x>6
First inequality. 5 y minus 10 greater than or equal to negative 3 x. Second inequality. y plus 5 greater than x. Third inequality. y plus 5 x greater than 6.
⎧⎩⎨⎪⎪5y−10≤−3xy+5<xy+5x<6
First inequality. 5 y minus 10 less than or equal to negative 3 x. Second inequality. y plus 5 less than x. Third inequality. y plus 5 x less than 6.
⎧⎩⎨⎪⎪3x+5y≤10−x+y>−5y−6>−5x
First inequality. 3 x plus 5 y less than or equal to 10. Second inequality. negative x plus y greater than negative 5. Third inequality. y minus 6 greater than negative 5 x.
⎧⎩⎨⎪⎪3x+5y≥10−x+y<−5y−6<−5x
First inequality. 3 x plus 5 y greater than or equal to 10. Second inequality. negative x plus y less than negative 5. Third inequality. y minus 6 less than negative 5 x.
A system of 3 inequalityar inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 10 to 10 in increments of 1. The vertical y-axis ranges from negative 10 to 10 in increments of 1. A solid inequality y equals negative fraction 3 over 5 end fraction x plus 2 passes through begin ordered pair 0 comma 2 end ordered pair and begin ordered pair 5 comma negative 1 end ordered pair. A dashed inequality y equals negative x minus 5 passing through begin ordered pair 0 comma negative 5 end ordered pair and begin ordered pair 5 comma 0 end ordered pair. A dashed inequality y equals negative 5 x plus 6 passes through begin ordered pair 0 comma 6 end ordered pair and begin ordered pair 2 comma negative 4 end ordered pair. The region below the solid inequality y equals negative fraction 3 over 5 end fraction x plus 2, above the dashed inequality y equals negative x minus 5 and above the dashed inequality inequality y equals negative 5 x plus 6 is shaded.
Responses
⎧⎩⎨⎪⎪5y−10≥−3xy+5>xy+5x>6
First inequality. 5 y minus 10 greater than or equal to negative 3 x. Second inequality. y plus 5 greater than x. Third inequality. y plus 5 x greater than 6.
⎧⎩⎨⎪⎪5y−10≤−3xy+5<xy+5x<6
First inequality. 5 y minus 10 less than or equal to negative 3 x. Second inequality. y plus 5 less than x. Third inequality. y plus 5 x less than 6.
⎧⎩⎨⎪⎪3x+5y≤10−x+y>−5y−6>−5x
First inequality. 3 x plus 5 y less than or equal to 10. Second inequality. negative x plus y greater than negative 5. Third inequality. y minus 6 greater than negative 5 x.
⎧⎩⎨⎪⎪3x+5y≥10−x+y<−5y−6<−5x
First inequality. 3 x plus 5 y greater than or equal to 10. Second inequality. negative x plus y less than negative 5. Third inequality. y minus 6 less than negative 5 x.
A system of 3 inequalityar inequalities graphed on a coordinate plane. The horizontal x-axis ranges from negative 10 to 10 in increments of 1. The vertical y-axis ranges from negative 10 to 10 in increments of 1. A solid inequality y equals negative fraction 3 over 5 end fraction x plus 2 passes through begin ordered pair 0 comma 2 end ordered pair and begin ordered pair 5 comma negative 1 end ordered pair. A dashed inequality y equals negative x minus 5 passing through begin ordered pair 0 comma negative 5 end ordered pair and begin ordered pair 5 comma 0 end ordered pair. A dashed inequality y equals negative 5 x plus 6 passes through begin ordered pair 0 comma 6 end ordered pair and begin ordered pair 2 comma negative 4 end ordered pair. The region below the solid inequality y equals negative fraction 3 over 5 end fraction x plus 2, above the dashed inequality y equals negative x minus 5 and above the dashed inequality inequality y equals negative 5 x plus 6 is shaded.
Answers
GPT-4o mini
To determine the correct system of inequalities that represents the given graph, we will analyze each provided system of inequalities, focusing on the graphical behavior described:
1. **First System**:
- \(5y - 10 \geq -3x\): This suggests a solid line and includes points on or above the line.
- \(y + 5 > x\): This suggests a dashed line and includes points above this line.
- \(y + 5x > 6\): This suggests a dashed line and includes points above this line as well.
2. **Second System**:
- \(5y - 10 \leq -3x\): This suggests a solid line and includes points below this line.
- \(y + 5 < x\): This suggests a dashed line and includes points below this line.
- \(y + 5x < 6\): This suggests a dashed line and includes points below this line.
3. **Third System**:
- \(3x + 5y \leq 10\): This suggests a solid line and includes points below this line.
- \(-x + y > -5\): This suggests a dashed line and includes points above this line.
- \(y - 6 > -5x\): This suggests a dashed line and includes points above this line.
4. **Fourth System**:
- \(3x + 5y \geq 10\): This suggests a solid line and includes points above this line.
- \(-x + y < -5\): This suggests a dashed line and includes points below this line.
- \(y - 6 < -5x\): This suggests a dashed line and includes points below this line.
**Graphical Analysis**:
- The solid line, which is a part of the system, indicates that the area below it is shaded.
- The dashed lines indicate that you are either above or below those lines based on the inequalities.
From your description:
- The solid line corresponds to a line of the form similar to \(y = -\frac{3}{5}x + 2\) which should signify a 'greater than or equal to' inequality.
- The dashed lines (where the region is shaded above) indicates that we are looking for inequalities reflecting points above these lines.
### Conclusion
Taking into account the shading behavior (solid line means greater than or equal, thus shading below; dashed means less than or above), it becomes clear that:
The correct system of inequalities that matches the described graph is:
**First System:**
\[
\begin{align*}
1. & \quad 5y - 10 \geq -3x \quad (\text{solid line}) \\
2. & \quad y + 5 > x \quad (\text{dashed line}) \\
3. & \quad y + 5x > 6 \quad (\text{dashed line}) \\
\end{align*}
\]
Thus, the answer is the **first system of inequalities.**
1. **First System**:
- \(5y - 10 \geq -3x\): This suggests a solid line and includes points on or above the line.
- \(y + 5 > x\): This suggests a dashed line and includes points above this line.
- \(y + 5x > 6\): This suggests a dashed line and includes points above this line as well.
2. **Second System**:
- \(5y - 10 \leq -3x\): This suggests a solid line and includes points below this line.
- \(y + 5 < x\): This suggests a dashed line and includes points below this line.
- \(y + 5x < 6\): This suggests a dashed line and includes points below this line.
3. **Third System**:
- \(3x + 5y \leq 10\): This suggests a solid line and includes points below this line.
- \(-x + y > -5\): This suggests a dashed line and includes points above this line.
- \(y - 6 > -5x\): This suggests a dashed line and includes points above this line.
4. **Fourth System**:
- \(3x + 5y \geq 10\): This suggests a solid line and includes points above this line.
- \(-x + y < -5\): This suggests a dashed line and includes points below this line.
- \(y - 6 < -5x\): This suggests a dashed line and includes points below this line.
**Graphical Analysis**:
- The solid line, which is a part of the system, indicates that the area below it is shaded.
- The dashed lines indicate that you are either above or below those lines based on the inequalities.
From your description:
- The solid line corresponds to a line of the form similar to \(y = -\frac{3}{5}x + 2\) which should signify a 'greater than or equal to' inequality.
- The dashed lines (where the region is shaded above) indicates that we are looking for inequalities reflecting points above these lines.
### Conclusion
Taking into account the shading behavior (solid line means greater than or equal, thus shading below; dashed means less than or above), it becomes clear that:
The correct system of inequalities that matches the described graph is:
**First System:**
\[
\begin{align*}
1. & \quad 5y - 10 \geq -3x \quad (\text{solid line}) \\
2. & \quad y + 5 > x \quad (\text{dashed line}) \\
3. & \quad y + 5x > 6 \quad (\text{dashed line}) \\
\end{align*}
\]
Thus, the answer is the **first system of inequalities.**