P/s : 8b-9a=31

Vì \(\frac{11}{7} > \frac{a}{b} > \frac{23}{29}\)

\(8 b - 9 a = 31\)(1)

\(\Rightarrow 9 a = 8 b - 31\)

\(a = \frac{8 b - 31}{9}\)vì \(a \in N\)

\(8 b - 31 \geq 9\)

\(\Leftrightarrow 8 b \geq 40 \Leftrightarrow b \geq 5\)

\(\Rightarrow \frac{11}{7} > \frac{8 b - 31}{9 b} > \frac{23}{29}\)

\(\Leftrightarrow \frac{11}{7} > \frac{8}{9} > \frac{23}{29}\)

Mà  \(7 > \frac{8}{9} - \frac{31}{9 b} < \frac{11}{7}\)

     \(\frac{8}{9} - \frac{11}{7} < \frac{31}{9 b}\)

      ...... \(\frac{- 43}{63} < \frac{31}{9 b}\)

\(\frac{- 43}{7} < \frac{31}{b}\)

\(\Leftrightarrow - 43 b < 31.7\)

\(b > \frac{31.7}{- 43} = \frac{- 217}{43}\)

\(\Rightarrow b \in N \Leftrightarrow b > 0\)

Mà \(\frac{8}{9} - \frac{31}{9 b} > \frac{23}{29} \Leftrightarrow \frac{8}{9} - \frac{23}{29} > \frac{31}{9 b}\)

\(\Leftrightarrow \frac{25}{261} > \frac{31}{9 b} \Rightarrow 25.9 b > 31.261\)

\(\Leftrightarrow b > \frac{31.261}{25.9} = \frac{899}{25} = 35 , 9\)

Vậy \(5 < b < \frac{899}{25}\)

\(\Rightarrow 5 < b < 35\)

Đến đây bạn lập bảng .