So King Solomon ruled over Israel,

1 Kings 3:28
When all Israel heard of the judgment the king had given, they stood in awe of him, for they saw that the wisdom of God was in him to administer justice.

1 Kings 4:2
and these were his chief officials: Azariah son of Zadok was the priest;

1 Kings 12:6
Then King Rehoboam consulted with the elders who had served his father Solomon during his lifetime. "How do you advise me to respond to these people?" he asked.

So king Solomon was king over all Israel.