Let's solve the inequalities: we have

and

Since the two inequality must be true at the same time (there is an "and" between the two), we should find a number that is, at the same time, greater than -6, and smaller than -8. But since -6 is greater than -8, a number greater than -6 is automatically greater than -8 as well. So, it is impossible for a number to be greater than -6 and smaller than -8.
If negative numbers confuse you, this example shows the same (impossible) logic: we can't ask for a number to be greater than 10, but smaller than 3.
As for the second exercise:


So, a number satisfies this system if it is smaller than 6 or greater than 1. This means that at least one of the conditions must be satisfied, and this is always the case:
- If we choose a number smaller than 1, the second condition is met
- If we choose a number between 1 and 6, both are met
- If we choose a number greater than 6, the first condition is met
So, whatever number we choose, at least one of the conditions will be true, and the logical "OR" will be satisfied.