To determine the solution set of the given graph of the dashed line and the area that is shaded above the line, we’ll first analyze the information provided.
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Identify the Equation of the Line: The points provided where the line intercepts the grid lines are:
- \((-10, -2)\)
- \((0, 0)\)
- \((10, 2)\)
From the two points \((0, 0)\) and \((10, 2)\), we can derive the equation of the line.
The slope (m) of the line can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{10 - 0} = \frac{2}{10} = \frac{1}{5} \]
Using the point-slope form of a linear equation \(y - y_1 = m(x - x_1)\) with the point \((0, 0)\), we get: \[ y - 0 = \frac{1}{5}(x - 0) \implies y = \frac{1}{5}x \]
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Determine the Inequality: Since the area above the line is shaded, this indicates that the inequality represented by the line is: \[ y > \frac{1}{5}x \]
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Finding the Whole Number Solutions: We need to find the whole number solutions \((x, y)\) that satisfy the inequality \(y > \frac{1}{5}x\).
To find suitable whole number pairs, we can test integer values of \(x\) and calculate \(y\) accordingly:
For \(x = -10\): \[ y > \frac{1}{5}(-10) = -2 \quad \Rightarrow \quad y \geq -1 \text{ (as y must be a whole number)} \] Solutions: \((-10, -1), (-10, 0), (-10, 1), (-10, 2), ..., (-10, 10)\)
For \(x = -9\): \[ y > \frac{1}{5}(-9) = -1.8 \quad \Rightarrow \quad y \geq -1 \] Solutions: \((-9, -1), (-9, 0), (-9, 1), ..., (-9, 10)\)
Continue for \(x = -8\) to \(x = 10\):
For \(x = -8\): \[ y > \frac{1}{5}(-8) = -1.6 \quad \Rightarrow \quad y \geq -1 \] Solutions: \((-8, -1), (-8, 0), (-8, 1), ..., (-8, 10)\)
For \(x = -7\): \[ y > \frac{1}{5}(-7) = -1.4 \quad \Rightarrow \quad y \geq -1 \] Solutions: \((-7, -1), (-7, 0), (-7, 1), ..., (-7, 10)\)
For \(x = -6\): \[ y > \frac{1}{5}(-6) = -1.2 \quad \Rightarrow \quad y \geq -1 \] Solutions: \((-6, -1), (-6, 0), (-6, 1), ..., (-6, 10)\)
For \(x = -5\): \[ y > \frac{1}{5}(-5) = -1 \quad \Rightarrow \quad y \geq 0 \] Solutions: \((-5, 0), (-5, 1), ..., (-5, 10)\)
Continue this process until \(x = 10\):
For \(x = 10\): \[ y > \frac{1}{5}(10) = 2 \quad \Rightarrow \quad y \geq 3 \] Solutions: \((10, 3), (10, 4), (10, 5), ..., (10, 10)\)
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Summarization of Whole Number Solutions: So, the whole number solutions \((x, y)\) in the area above the line includes:
- For \(x\) values from -10 to 10.
- Whole \(y\) values such that \(y > \frac{1}{5}x\).
In conclusion, every integer point in the shaded region above the dashed line forms a part of the solution set for the given inequality \(y > \frac{1}{5}x\). The exact values of \(y\) depend on the specific \(x\) as described above.
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