\(\frac{2}{3}\left(x+1\right)< \frac{2x+1}{5}\)
\(\Leftrightarrow\frac{5.2\left(x+1\right)}{3.5}< \frac{3.\left(2x+1\right)}{5.3}\)
\(\Leftrightarrow10\left(x+1\right)< 3\left(2x+1\right)\)
\(\Leftrightarrow10x+10< 6x+3\)
\(\Leftrightarrow4x< -7\)
\(\Leftrightarrow x< -\frac{7}{4}\)