To determine which of the given points is a solution to the linear inequality \( y > \frac{1}{9}x - \frac{2}{9} \), we will evaluate the inequality at \( x = 0 \) since all provided points have \( x = 0 \).
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Substitute \( x = 0 \) into the inequality: \[ y > \frac{1}{9}(0) - \frac{2}{9} \] This simplifies to: \[ y > -\frac{2}{9} \]
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Now we will evaluate each of the provided points to see if the \( y \)-value satisfies the inequality \( y > -\frac{2}{9} \).
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For the point \( (0, -49) \): \[ -49 > -\frac{2}{9} \quad \text{(False)} \]
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For the point \( (0, -\frac{4}{9}) \): \[ -\frac{4}{9} > -\frac{2}{9} \quad \text{(False)} \]
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For the point \( (0, -29) \): \[ -29 > -\frac{2}{9} \quad \text{(False)} \]
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For the point \( (0, -\frac{2}{9}) \): \[ -\frac{2}{9} > -\frac{2}{9} \quad \text{(False)} \]
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For the point \( (0, -13) \): \[ -13 > -\frac{2}{9} \quad \text{(False)} \]
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For the point \( (0, - \frac{1}{3}) \): \[ -\frac{1}{3} > -\frac{2}{9} \] To compare these values, convert \(-\frac{1}{3}\) to have a common denominator of 9: \[ -\frac{1}{3} = -\frac{3}{9} \] Thus: \[ -\frac{3}{9} > -\frac{2}{9} \quad \text{(True)} \]
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For the point \( (0, -19) \): \[ -19 > -\frac{2}{9} \quad \text{(False)} \]
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The point that satisfies the inequality \( y > -\frac{2}{9} \) is \( (0, -\frac{1}{3}) \).
Therefore, the solution to the inequality is (0, −Start Fraction 1 over 3 End Fraction).