To determine which point is a solution to the system of inequalities, we need to evaluate each point against the inequalities given:
- Inequality 1: \( y \leq -\frac{1}{5}x - 8 \)
- Inequality 2: \( y \geq 5x + 5 \)
Let's evaluate each point:
Point (-5, 3)
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Evaluate \( y \leq -\frac{1}{5}x - 8 \): \[ 3 \leq -\frac{1}{5}(-5) - 8 \] \[ 3 \leq 1 - 8 \implies 3 \leq -7 \quad (\text{False}) \]
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Evaluate \( y \geq 5x + 5 \): \[ 3 \geq 5(-5) + 5 \] \[ 3 \geq -25 + 5 \implies 3 \geq -20 \quad (\text{True}) \]
Point (-10, 5)
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Evaluate \( y \leq -\frac{1}{5}x - 8 \): \[ 5 \leq -\frac{1}{5}(-10) - 8 \] \[ 5 \leq 2 - 8 \implies 5 \leq -6 \quad (\text{False}) \]
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Evaluate \( y \geq 5x + 5 \): \[ 5 \geq 5(-10) + 5 \] \[ 5 \geq -50 + 5 \implies 5 \geq -45 \quad (\text{True}) \]
Point (-10, -6)
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Evaluate \( y \leq -\frac{1}{5}x - 8 \): \[ -6 \leq -\frac{1}{5}(-10) - 8 \] \[ -6 \leq 2 - 8 \implies -6 \leq -6 \quad (\text{True}) \]
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Evaluate \( y \geq 5x + 5 \): \[ -6 \geq 5(-10) + 5 \] \[ -6 \geq -50 + 5 \implies -6 \geq -45 \quad (\text{True}) \]
Point (-5, 2)
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Evaluate \( y \leq -\frac{1}{5}x - 8 \): \[ 2 \leq -\frac{1}{5}(-5) - 8 \] \[ 2 \leq 1 - 8 \implies 2 \leq -7 \quad (\text{False}) \]
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Evaluate \( y \geq 5x + 5 \): \[ 2 \geq 5(-5) + 5 \] \[ 2 \geq -25 + 5 \implies 2 \geq -20 \quad (\text{True}) \]
Summary of Results:
- Point (-5, 3): True for the second, false for the first.
- Point (-10, 5): True for the second, false for the first.
- Point (-10, -6): True for both inequalities.
- Point (-5, 2): True for the second, false for the first.
Conclusion:
The only point that satisfies both inequalities is (-10, -6). Thus, the correct answer is: (-10, -6).