\( \left. \begin{array} { l } { x \in Z } \\ { \frac { 2 } { x - 6 } < \frac { - 5 } { 3 } } \\ { \Rightarrow x = ? } \end{array} \right. \)

To solve the inequality \( \frac{2}{x - 6} < \frac{-5}{3} \), we first cross-multiply to eliminate the fractions, but be cautious since it involves the sign of the inequality! We rearrange this to get \( 2 \cdot 3 < -5(x - 6) \), leading to \( 6 < -5x + 30 \). Now, simplify this to find \( 5x < 30 - 6 \), which equals \( 5x < 24 \). Dividing both sides by 5 gives us \( x < \frac{24}{5} \) or \( x < 4.8 \). Since \( x \) must be an integer, the possible values for \( x \) are \( x \leq 4 \). However, remember that the expression \( \frac{2}{x - 6} \) is undefined at \( x = 6 \). So, the final solution is \( x \in \mathbb{Z} \) such that \( x < 5 \) (or specifically, \( x = 4, 3, 2, 1, 0, -1, \ldots \)). The integer solutions are \( x = 4, 3, 2, 1, 0, -1, -2, \ldots\).