We can rewrite the two sets as
[tex]x+1<4 \iff x<3,\quad x-8>-7 \iff x>1[/tex]
So, the first set is composed by all numbers smaller then 3, and the second is composed by all numbers greater than 1.
The intersection of two sets is composed by all the elements appearing in both sets, so
[tex]\{x<3\}\cap \{x>1\}[/tex]
Is the set composed by all number that are less than 3 and more than 1, at the same time.
In other words, you're simply asking
[tex]\{1<x<3\}[/tex]
