To determine the equations that represent the inequalities in the system based on the given descriptions of the lines:
Line 1:
- The dashed line going through the points \((-1, 0)\) and \((0, 2)\) can be analyzed to find its slope:
- The slope (m) between these two points is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{0 - (-1)} = \frac{2}{1} = 2 \]
- The line equation in slope-intercept form \(y = mx + b\) can be derived. With point \((0, 2)\) giving \(b = 2\): \[ y = 2x + 2 \]
- The inequality is dashed, which indicates that points on the line are not included, and given that everything to the right is shaded, the inequality is: \[ y > 2x + 2 \]
Line 2:
- The solid line going through \((0, -1)\) and \((1, 1)\) can also be analyzed for its slope:
- The slope (m) is given by: \[ m = \frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2 \]
- The line equation is found similarly. With point \((0, -1)\) giving \(b = -1\): \[ y = 2x - 1 \]
- The line is solid, indicating that points on the line are included. Since areas to the left of the line are shaded, the inequality is: \[ y \leq 2x - 1 \]
System of Inequalities:
Thus, the system of inequalities can be expressed as:
- \(y > 2x + 2\) (for the dashed line)
- \(y \leq 2x - 1\) (for the solid line)
Point as Solution:
Now we need to determine which points satisfy both inequalities. Let's check a few points:
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Testing the Point (0, 0):
- For \(y > 2x + 2\): \[ 0 > 2(0) + 2 \implies 0 > 2 \quad \text{(false)} \]
- This point does not satisfy the first inequality.
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Testing the Point (1, 0):
- For \(y > 2x + 2\): \[ 0 > 2(1) + 2 \implies 0 > 4 \quad \text{(false)} \]
- This point does not satisfy the first inequality.
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Testing the Point (-1, 1):
- For \(y > 2x + 2\): \[ 1 > 2(-1) + 2 \implies 1 > 0 \quad \text{(true)} \]
- For \(y \leq 2x - 1\): \[ 1 \leq 2(-1) - 1 \implies 1 \leq -3 \quad \text{(false)} \]
- This point does not satisfy the second inequality.
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Testing the Point (-1, -1):
- For \(y > 2x + 2\): \[ -1 > 2(-1) + 2 \implies -1 > 0 \quad \text{(false)} \]
- This point does not satisfy the first inequality.
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Testing the Point (0, 1):
- For \(y > 2x + 2\): \[ 1 > 2(0) + 2 \implies 1 > 2 \quad \text{(false)} \]
- This point does not satisfy the first inequality.
After testing several points, you can find suitable candidates that lie on the left side of the second line and above the first line. Typically, you can check points that are close to the lines or intersections.
Conclusion:
The inequalities of the system are:
- \(y > 2x + 2\)
- \(y \leq 2x - 1\)
And a point that could serve as a solution should be identified through testing or by observing intersection points of the lines if drawn accurately. For exact candidates, I would suggest checking the points of intersection or points slightly above the dashed line and to the left of the solid line based on the graphs provided.