To solve the system of inequalities by graphing and identify the solution region, follow these steps:
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Graph each inequality:
For the first inequality, \( y > -\frac{1}{2}x + 3 \):
- This line has a slope of \(-\frac{1}{2}\) and a y-intercept of \(3\).
- To graph it, plot the y-intercept (0, 3) and use the slope to find another point: from (0, 3), go down 1 unit and right 2 units to reach (2, 2).
- Draw a dotted line since the inequality is strict (greater than, not greater than or equal to), indicating that points on the line are not included in the solution.
For the second inequality, \( y \leq x - 15 \):
- This line has a slope of \(1\) and a y-intercept of \(-15\).
- To graph it, plot the y-intercept (0, -15) and use the slope to find another point: from (0, -15), go up 1 unit and right 1 unit to reach (1, -14).
- Draw a solid line since the inequality includes equal to (less than or equal to), indicating that points on the line are included in the solution.
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Shading the regions:
- For the first inequality \( y > -\frac{1}{2}x + 3 \): shade the region above the dotted line.
- For the second inequality \( y \leq x - 15 \): shade the region below (or including) the solid line.
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Finding the solution region:
- The solution to the system of inequalities will be the area where the shaded regions overlap.
Thus, the region that represents the solution to the system of inequalities is where the area above the dotted line (from the first inequality) and below or on the solid line (from the second inequality) intersect.
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