How does one explain to a close relative that living together without marriage is a sin? This relative is not WELS. In fact, they are Catholic.
Your concern for your relative is commendable. You certainly want to speak the truth in love (Ephesians 4:15) to your relative.
You can explain that when two people of the opposite sex live together without the benefit of marriage, they put themselves into tempting situations where they can break the sixth commandment. The Bible tells Christians to “flee from sexual immorality” (1 Corinthians 6:18). God explains that he “will judge the adulterer and all the sexually immoral” (Hebrews 13:4).
If a man and a woman living together without being married claim to be refraining from sexual activity, they are still in a position of causing offense to others (Matthew 18:6-7) and emboldening them to sin. In addition, their lives are not in line with what God says: that “among you, there must not be even a hint of sexual immorality” (Ephesians 5:3).
Besides speaking to your relative, continue to include him or her in your prayers. Pray that God will change hearts and attitudes and lives. God bless your conversations.