A system of congruences

First two. $x\equiv5\pmod8$, $x\equiv3\pmod9$. Try $x=5+8k$: $5+8k\equiv3\pmod9$, $8k\equiv-2\equiv7\pmod9$. $8^{-1}\equiv8\pmod9$ (since $8\cdot8=64\equiv1$), so $k\equiv7\cdot8=56\equiv2\pmod9$. $x=5+8(2+9m)=21+72m$.

Combine with third. $21+72m\equiv4\pmod7$. $21\equiv0$, $72\equiv2\pmod7$, so $2m\equiv4\pmod7$, $m\equiv2\pmod7$. $x=21+72(2+7n)=165+504n$.

Conclusion. $x\equiv165\pmod{504}$.

Check. $165\equiv5\pmod8$, $165\equiv3\pmod9$, $165\equiv4\pmod7$. Correct.