5. The solution set of the equation $\left[\frac{x+1}{10}\right]=\left[\frac{x-1}{5}\right]$ is
Solution: Let $\left[\frac{x+1}{10}\right]=\left[\frac{x-1}{5}\right]=m, m \in Z$,
$\therefore m \leq \frac{x+1}{10}<m+1$ and $m \leq \frac{x-1}{5}<m+1$
$\therefore 10 m-1 \leq x<10 m+9$ and $5 m+1 \leq x<6 m+6$
Since $10 m+9-(10 m-1)=10,5 m+6-(5 m+1)=5$
Discussion as follows
(1) When $5 m+6 \leq 10 m-1$, i.e., $m \geq \frac{7}{5}$, at this time $m=2,3, \cdots$ the equation has no solution (as shown in Figure 5-1)
Figure 5-1
(2) When $5 m+1 \leq 10 m-1<5 m+6 \leq 10 m+9$, i.e., $\frac{2}{5} \leq m<\frac{7}{5}$, thus $m=1$
(2) When $5 m+1 \leq 10 m-1<5 m+6 \leq 10 m+9$, i.e., $\frac{2}{5} \leq m<\frac{7}{5}$, thus $m=1$. The solution set of the equation is $[9,11$ ) (as shown in Figure 5-2)
Figure 5-2
(3) When $10 m-1<5 m+1<5 m+6 \leq 10 m+9$, i.e., $-\frac{3}{5} \leq m<\frac{2}{5}$, thus $m=0$. The solution set of the equation is $[1,6$ ) (as shown in Figure 5-3)
Figure 5-3
(4) When
Figure 5-3
$10 m-1<5 m+1<10 m+9<5 m+6$, i.e., $-\frac{8}{5} \leq m<-\frac{3}{5}$, thus $m=-1$. The solution set of the equation is $[-4,-1$ ) (as shown in Figure 5-4)
(5) When $10 m+9 \leq 5 m+1$, i.e., $m \leq-\frac{8}{5}$, at this time $m=-2,-3, \cdots$ the equation has no solution (as shown in Figure 5-5)

In summary, the solution set of the original equation is $[-4,-1) \cup[1,6) \cup[9,11)$.