Suppose $x\ge 4$. This means neither $x+1$ nor $x-4$ are negative, so the inequality becomes $$x+1+x-4> 7$$, which simplifies to $2x\ge10$, and $x\ge 5$. So the inequality holds when $x> 5$.
Suppose $-1\le x<4$, so $x+1\ge 0$, but $x-4<0$. The inequality becomes
$$x+1-x+4> 7$$
which simplifies to $5> 7$ and that's obviously not true.
Suppose $x<-1$. This means both $x+1$ and $x-4$ are negative, so the inequality becomes:
$$-x-1-x+4> 7$$
Which simplifies to $-2x> 4$, so $x< -2$.
We conclude the inequality holds for all $x> 5$ and all $x< -2$