To determine which of the points are solutions to the system of inequalities \( y < x + 8 \) and \( y > -\frac{1}{2}x \), we will evaluate each point against the two inequalities.
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Point (–2, 0):
For \( y < x + 8 \): \[ 0 < -2 + 8 \implies 0 < 6 \quad \text{(True)} \] For \( y > -\frac{1}{2}x \): \[ 0 > -\frac{1}{2}(-2) \implies 0 > 1 \quad \text{(False)} \]This point does not satisfy both inequalities.
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Point (10, –1):
For \( y < x + 8 \): \[ -1 < 10 + 8 \implies -1 < 18 \quad \text{(True)} \] For \( y > -\frac{1}{2}x \): \[ -1 > -\frac{1}{2}(10) \implies -1 > -5 \quad \text{(True)} \]This point satisfies both inequalities.
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Point (2, 4):
For \( y < x + 8 \): \[ 4 < 2 + 8 \implies 4 < 10 \quad \text{(True)} \] For \( y > -\frac{1}{2}x \): \[ 4 > -\frac{1}{2}(2) \implies 4 > -1 \quad \text{(True)} \]This point satisfies both inequalities.
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Point (0, –10):
For \( y < x + 8 \): \[ -10 < 0 + 8 \implies -10 < 8 \quad \text{(True)} \] For \( y > -\frac{1}{2}x \): \[ -10 > -\frac{1}{2}(0) \implies -10 > 0 \quad \text{(False)} \]This point does not satisfy both inequalities.
Summary of Results
The solutions that satisfy the system of inequalities are:
- (10, –1)
- (2, 4)
Thus, the selected points that are solutions to the system of inequalities are (10, –1) and (2, 4).