Does God make a difference between Jews and Gentiles or rich and poor in giving righteousness?

No. Scripture teaches that the righteousness of God, which is by the faith of Jesus Christ, is unto all and upon all who believe. There is no difference between Jew and Gentile, black and white, rich and poor; God’s righteousness is imputed to and covers every believer.

Les explains that Paul often uses contemporary illustrations, such as the Roman soldier who carried his weapon in his right hand and shield in his left. Paul uses that analogy but replaces sword and shield with righteousness in both hands. He then points back to Romans chapter 3, verse 22, showing that this righteousness is not only for a few believers but for every believer. The righteousness of God, which is by the faith of Jesus Christ, is unto all and upon all who believe, for there is no difference between Jew and Gentile, black and white, rich and poor. The righteousness of God has been imputed and covers every believer.

Romans 3:22

Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference:

What does God's righteousness do for every believer?

God's righteousness is imputed to and covers every believer, without distinction between Jew and Gentile, rich and poor, or black and white. It comes by the faith of Jesus Christ and serves as the believer's armor, held in both hands for the battle.