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Sagot :
Claro, vamos a determinar la relación de orden entre cada par de fracciones dadas. Para comparar fracciones, podemos convertirlas a una forma decimal o encontrar un común denominador. Aquí están los resultados de la comparación:
1. Comparando [tex]$\frac{6}{7}$[/tex] y [tex]$\frac{9}{11}$[/tex]:
[tex]\[ \frac{6}{7} > \frac{9}{11} \][/tex]
Porque [tex]\(0.8571 > 0.8181\)[/tex].
2. Comparando [tex]$\frac{3}{5}$[/tex] y [tex]$\frac{3}{9}$[/tex]:
[tex]\[ \frac{3}{5} > \frac{3}{9} \][/tex]
Porque [tex]\(0.6 > 0.3333\)[/tex].
3. Comparando [tex]$\frac{12}{15}$[/tex] y [tex]$\frac{14}{15}$[/tex]:
[tex]\[ \frac{12}{15} < \frac{14}{15} \][/tex]
Porque [tex]\(0.8 < 0.9333\)[/tex].
4. Comparando [tex]$\frac{-2}{3}$[/tex] y [tex]$\frac{-5}{8}$[/tex]:
[tex]\[ \frac{-2}{3} < \frac{-5}{8} \][/tex]
Porque [tex]\( -0.6667 < -0.625 \)[/tex].
5. Comparando [tex]$\frac{-4}{5}$[/tex] y [tex]$\frac{-8}{10}$[/tex]:
[tex]\[ \frac{-4}{5} = \frac{-8}{10} \][/tex]
Porque [tex]\( -0.8 = -0.8 \)[/tex].
6. Comparando [tex]$\frac{13}{21}$[/tex] y [tex]$\frac{27}{45}$[/tex]:
[tex]\[ \frac{13}{21} > \frac{27}{45} \][/tex]
Porque [tex]\( 0.6190 > 0.6 \)[/tex].
Por lo tanto, las relaciones de orden correctas para cada par de fracciones son:
[tex]\[ \frac{6}{7} > \frac{9}{11}, \quad \frac{3}{5} > \frac{3}{9}, \quad \frac{12}{15} < \frac{14}{15}, \quad \frac{-2}{3} < \frac{-5}{8}, \quad \frac{-4}{5} = \frac{-8}{10}, \quad \frac{13}{21} > \frac{27}{45} \][/tex]
Resumiendo en el formato solicitado, tenemos:
[tex]\[ \begin{tabular}{cc|cc} \frac{6}{7} & \frac{9}{11} & \frac{3}{5} & \frac{3}{9} \\ \hline > & > & < \\ \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{cc|cc} \frac{12}{15} & \frac{14}{15} & \frac{-2}{3} & \frac{-5}{8} \\ \hline < & \quad< & = \\ \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{cc} \frac{-4}{5} & \frac{-8}{10} \\ \hline = \\ \end{tabular} \][/tex]
\]
\begin{tabular}{cc}
\frac{13}{21} & \frac{27}{45} \\
\hline
> \\
\end{tabular}
1. Comparando [tex]$\frac{6}{7}$[/tex] y [tex]$\frac{9}{11}$[/tex]:
[tex]\[ \frac{6}{7} > \frac{9}{11} \][/tex]
Porque [tex]\(0.8571 > 0.8181\)[/tex].
2. Comparando [tex]$\frac{3}{5}$[/tex] y [tex]$\frac{3}{9}$[/tex]:
[tex]\[ \frac{3}{5} > \frac{3}{9} \][/tex]
Porque [tex]\(0.6 > 0.3333\)[/tex].
3. Comparando [tex]$\frac{12}{15}$[/tex] y [tex]$\frac{14}{15}$[/tex]:
[tex]\[ \frac{12}{15} < \frac{14}{15} \][/tex]
Porque [tex]\(0.8 < 0.9333\)[/tex].
4. Comparando [tex]$\frac{-2}{3}$[/tex] y [tex]$\frac{-5}{8}$[/tex]:
[tex]\[ \frac{-2}{3} < \frac{-5}{8} \][/tex]
Porque [tex]\( -0.6667 < -0.625 \)[/tex].
5. Comparando [tex]$\frac{-4}{5}$[/tex] y [tex]$\frac{-8}{10}$[/tex]:
[tex]\[ \frac{-4}{5} = \frac{-8}{10} \][/tex]
Porque [tex]\( -0.8 = -0.8 \)[/tex].
6. Comparando [tex]$\frac{13}{21}$[/tex] y [tex]$\frac{27}{45}$[/tex]:
[tex]\[ \frac{13}{21} > \frac{27}{45} \][/tex]
Porque [tex]\( 0.6190 > 0.6 \)[/tex].
Por lo tanto, las relaciones de orden correctas para cada par de fracciones son:
[tex]\[ \frac{6}{7} > \frac{9}{11}, \quad \frac{3}{5} > \frac{3}{9}, \quad \frac{12}{15} < \frac{14}{15}, \quad \frac{-2}{3} < \frac{-5}{8}, \quad \frac{-4}{5} = \frac{-8}{10}, \quad \frac{13}{21} > \frac{27}{45} \][/tex]
Resumiendo en el formato solicitado, tenemos:
[tex]\[ \begin{tabular}{cc|cc} \frac{6}{7} & \frac{9}{11} & \frac{3}{5} & \frac{3}{9} \\ \hline > & > & < \\ \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{cc|cc} \frac{12}{15} & \frac{14}{15} & \frac{-2}{3} & \frac{-5}{8} \\ \hline < & \quad< & = \\ \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{cc} \frac{-4}{5} & \frac{-8}{10} \\ \hline = \\ \end{tabular} \][/tex]
\]
\begin{tabular}{cc}
\frac{13}{21} & \frac{27}{45} \\
\hline
> \\
\end{tabular}
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