To determine which of the given points is a solution to the linear inequality \( y > \frac{1}{9}x - \frac{2}{9} \), we can substitute each point into the inequality and check if the inequality holds true.
The inequality in question is: \[ y > \frac{1}{9}x - \frac{2}{9} \]
Let's evaluate each point:
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Point (0, -4/9): \[ y = -\frac{4}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{4}{9} > -\frac{2}{9} \] This is false.
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Point (0, -1/9): \[ y = -\frac{1}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{1}{9} > -\frac{2}{9} \] This is true.
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Point (0, -2/9): \[ y = -\frac{2}{9}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{2}{9} > -\frac{2}{9} \] This is false (because it’s not greater).
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Point (0, -1/3): \[ y = -\frac{1}{3}, \quad x = 0 \Rightarrow y > \frac{1}{9}(0) - \frac{2}{9} \Rightarrow -\frac{1}{3} > -\frac{2}{9} \] To compare -1/3 and -2/9, convert -1/3 to a fraction with common denominator 9: \[ -\frac{1}{3} = -\frac{3}{9} \] Then we check: \[ -\frac{3}{9} > -\frac{2}{9} \] This is false.
The only point that satisfies the inequality is (0, -1/9).