The open point plotted at \( 6 \frac{2}{3} \) (which is equal to \( \frac{20}{3} \) or approximately 6.67) on the number line indicates that the solution to the inequality starts at this point and extends to the right towards positive infinity. This suggests that the correct inequality is one where \( x \) must be greater than \( 6 \frac{2}{3} \).
Let's analyze each provided response to check which one corresponds to this interval.
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\( 15x > 113 \): To solve for \( x \): \[ x > \frac{113}{15} \approx 7.53 \] (This is greater than \( 6 \frac{2}{3} \))
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\( \frac{1}{5}x > 1 \frac{1}{3} \): To solve for \( x \): \[ \frac{1}{5}x > \frac{4}{3} \implies x > \frac{4}{3} \cdot 5 = \frac{20}{3} \ \] (This corresponds to exactly \( 6 \frac{2}{3} \) since \( \frac{20}{3} = 6 \frac{2}{3} \))
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\( 13x > 225 \): To solve for \( x \): \[ x > \frac{225}{13} \approx 17.31 \] (This is much greater)
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\( \frac{1}{2}x > 3 \frac{1}{4} \): To solve for \( x \): \[ \frac{1}{2}x > \frac{13}{4} \implies x > \frac{13}{4} \cdot 2 = \frac{26}{4} = 6.5 \] (This is less than \( 6 \frac{2}{3} \))
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\( 14x > 212 \): To solve for \( x \): \[ x > \frac{212}{14} \approx 15.14 \] (This is much greater)
Thus, among the options given, the only inequality that represents the solution \( x > 6 \frac{2}{3} \) is:
\( \frac{1}{5}x > 1 \frac{1}{3} \).