Let's analyze each of the given inequalities one by one to determine if it is true or false.
### Inequality 1: [tex]\(\frac{7}{8} < \frac{6}{7}\)[/tex]
We need to compare the fractions [tex]\(\frac{7}{8}\)[/tex] and [tex]\(\frac{6}{7}\)[/tex].
Comparing [tex]\(\frac{7}{8}\)[/tex] to [tex]\(\frac{6}{7}\)[/tex]:
[tex]\[
\frac{7}{8} \approx 0.875 \quad \text{and} \quad \frac{6}{7} \approx 0.857
\][/tex]
Since [tex]\(0.875 > 0.857\)[/tex], we find that:
[tex]\[
\frac{7}{8} > \frac{6}{7}
\][/tex]
Therefore, the inequality [tex]\(\frac{7}{8} < \frac{6}{7}\)[/tex] is False.
### Inequality 2: [tex]\(\frac{9}{10} > \frac{11}{12}\)[/tex]
We need to compare the fractions [tex]\(\frac{9}{10}\)[/tex] and [tex]\(\frac{11}{12}\)[/tex].
Comparing [tex]\(\frac{9}{10}\)[/tex] to [tex]\(\frac{11}{12}\)[/tex]:
[tex]\[
\frac{9}{10} = 0.9 \quad \text{and} \quad \frac{11}{12} \approx 0.917
\][/tex]
Since [tex]\(0.9 < 0.917\)[/tex], we see that:
[tex]\[
\frac{9}{10} < \frac{11}{12}
\][/tex]
Therefore, the inequality [tex]\(\frac{9}{10} > \frac{11}{12}\)[/tex] is False.
### Inequality 3: [tex]\(\frac{2}{3} > \frac{9}{13}\)[/tex]
We need to compare the fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{9}{13}\)[/tex].
Comparing [tex]\(\frac{2}{3}\)[/tex] to [tex]\(\frac{9}{13}\)[/tex]:
[tex]\[
\frac{2}{3} \approx 0.667 \quad \text{and} \quad \frac{9}{13} \approx 0.692
\][/tex]
Since [tex]\(0.667 < 0.692\)[/tex], we find that:
[tex]\[
\frac{2}{3} < \frac{9}{13}
\][/tex]
Therefore, the inequality [tex]\(\frac{2}{3} > \frac{9}{13}\)[/tex] is False.
### Inequality 4: [tex]\(\frac{1}{4} < \frac{2}{7}\)[/tex]
We need to compare the fractions [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
Comparing [tex]\(\frac{1}{4}\)[/tex] to [tex]\(\frac{2}{7}\)[/tex]:
[tex]\[
\frac{1}{4} = 0.25 \quad \text{and} \quad \frac{2}{7} \approx 0.286
\][/tex]
Since [tex]\(0.25 < 0.286\)[/tex], we see that:
[tex]\[
\frac{1}{4} < \frac{2}{7}
\][/tex]
Therefore, the inequality [tex]\(\frac{1}{4} < \frac{2}{7}\)[/tex] is True.
### Conclusion
After analyzing all four inequalities, we conclude that the only true inequality is:
[tex]\(\frac{1}{4} < \frac{2}{7}\)[/tex]
