Question attatched
None of the intervals can be unbounded, right or left rays. Otherwise there is a counter-example.
Name the interval with the least left end point $\displaystyle I_1$. There are at most six intervals that may share a point with it. So we have at least 43 which have no point common with $\displaystyle I_1$. Of those, name the interval with the least left end point $\displaystyle I_2$. Again are at most six intervals that may share a point with $\displaystyle I_2$, leaving 36 that have no point either of the two named. We continue getting $\displaystyle I_3$ from 28 and so fourth until we get $\displaystyle I_8$ from 8.
Now you have 8 intervals that are pairwise disjoint.