In the first line it is said that 6/-7 is a negative rational number but the line after that I marked is saying that rational numbers with negative integers are eliminated. How is that possible.

Dear Student,
The explanation says that $\frac{9}{-1}$ is a negative rational number, but generally we do not write minus sign in the denominator, therefore, we do not represent -9 as $\frac{9}{-1}$. 
Hope this helps you! 

Regards