"The first to plead his case seems right, until another comes and examines him"
This is the Hebrew word rishon, which means first or chief or former.
plead his case
This is the Hebrew word rib, which means strife, dispute, controversy.
The idea is that someone is presenting a disputed situation. The first person who tells his side of the story makes it seem like his point of view is right.
Solomon is trying to get us to not jump to conclusions but to wait until the other side is presented and other facts are brought out. Everyone can make their side of a disputed situation seem like they are right but wait and see what the other side has to say. DON'T JUMP TO CONCLUSIONS.
This is the Hebrew word saddiq, which means just, righteous, right.
This is the Hebrew word rea, which means friend, companion; also it can be used for opponent. But it often has a very close relational meaning. It is interesting that often those we have strife and disagreements with are those who we were very close to before.
This is the Hebrew word chaqar, which means to search, to examine, to investigate. The idea here is that a more thorough investigation or questioning by their accuser can often uncover other perspectives and get closer to the truth.
Solomon is trying to get us to remember that when we are hearing one side of a story, there is another side. The first person is not meaning to lie, but they are presenting their story in the very best light possible and they are leaving out any facts that do not help their version. Take this as a general rule. This is true with children explaining something to their parents. It is true with employees giving a report to their boss. It is true with married couples explaining or arguing for their position. Do not get upset at this; just realize that this is going to happen and you must wait to make a decision. You must wait before your emotions are allowed to engage until you have heard the other perspective. WAIT UNTIL YOU HAVE HEARD THE OTHER SIDE.