Let's evaluate each of the inequalities one by one:
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One-half less than one-third: \[ \frac{1}{2} \text{ vs } \frac{1}{3} \] Since \(\frac{1}{2} = 0.5\) and \(\frac{1}{3} \approx 0.333\), we have: \[ \frac{1}{2} > \frac{1}{3} \quad \text{(This inequality is false)} \]
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Three-fourths greater than two-thirds: \[ \frac{3}{4} \text{ vs } \frac{2}{3} \] Converting both fractions to a common denominator:
- \(\frac{3}{4} = \frac{9}{12}\)
- \(\frac{2}{3} = \frac{8}{12}\) Since \(\frac{9}{12} > \frac{8}{12}\): \[ \frac{3}{4} > \frac{2}{3} \quad \text{(This inequality is true)} \]
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Negative one-fourth less than negative two-thirds: \[ -\frac{1}{4} \text{ vs } -\frac{2}{3} \] Converting both fractions to a common denominator:
- \(-\frac{1}{4} = -\frac{3}{12}\)
- \(-\frac{2}{3} = -\frac{8}{12}\) Since \(-\frac{3}{12} > -\frac{8}{12}\): \[ -\frac{1}{4} > -\frac{2}{3} \quad \text{(This inequality is false)} \]
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Negative 1 greater than three-fourths: \[ -1 \text{ vs } \frac{3}{4} \] Since \(-1 < \frac{3}{4}\): \[ -1 < \frac{3}{4} \quad \text{(This inequality is false)} \]
In summary, the only true inequality from the options given is: \[ \frac{3}{4} > \frac{2}{3} \]