To determine which of the given inequalities is true, let's analyze each one by converting the fractions to decimals and then comparing them.
### Option A:
[tex]\[
\frac{3}{4} \quad \text{and} \quad \frac{5}{7}
\][/tex]
Convert each fraction to a decimal:
[tex]\[
\frac{3}{4} = 0.75
\][/tex]
[tex]\[
\frac{5}{7} \approx 0.714
\][/tex]
Compare:
[tex]\[
0.75 \not< 0.714
\][/tex]
So, [tex]\(\frac{3}{4} < \frac{5}{7}\)[/tex] is false.
### Option B:
[tex]\[
\frac{2}{3} \quad \text{and} \quad \frac{5}{6}
\][/tex]
Convert each fraction to a decimal:
[tex]\[
\frac{2}{3} \approx 0.667
\][/tex]
[tex]\[
\frac{5}{6} \approx 0.833
\][/tex]
Compare:
[tex]\[
0.667 \not> 0.833
\][/tex]
So, [tex]\(\frac{2}{3} > \frac{5}{6}\)[/tex] is false.
### Option C:
[tex]\[
\frac{5}{8} \quad \text{and} \quad \frac{6}{10}
\][/tex]
Convert each fraction to a decimal:
[tex]\[
\frac{5}{8} = 0.625
\][/tex]
[tex]\[
\frac{6}{10} = 0.6
\][/tex]
Compare:
[tex]\[
0.625 > 0.6
\][/tex]
So, [tex]\(\frac{5}{8} > \frac{6}{10}\)[/tex] is true.
### Option D:
[tex]\[
\frac{4}{5} \quad \text{and} \quad \frac{2}{9}
\][/tex]
Convert each fraction to a decimal:
[tex]\[
\frac{4}{5} = 0.8
\][/tex]
[tex]\[
\frac{2}{9} \approx 0.222
\][/tex]
Compare:
[tex]\[
0.8 \not< 0.222
\][/tex]
So, [tex]\(\frac{4}{5} < \frac{2}{9}\)[/tex] is false.
### Conclusion:
The true inequality is:
[tex]\[
\boxed{\frac{5}{8} > \frac{6}{10}}
\][/tex]
