In general $F_{X}(x)=P(X \le x)$ and one uses this fact to compute probabilities of intervals. In the case of an interval $x>c$ we can use the fact that $P(X>c)=1-P(X \le c)$, which is $1-F(c)$ by definition. (I dropped the subscript $X$ on the function F.)
Also in general to find $P(a one applies the formula $F(b-)-F(a).$ The extra tag of $-$ after the $b$ indicate limit from the left. The left limit at $b$ is necessary only if $F$ is discontinuous at $b$.
There are plenty of other possible forms of intervals, all treated in a similar way. But if you notice that your cumulative is continuous, as in your example, you can safely just plug in endpoints and subtract.