Like how a point guard (also called a “ball-handler”) handles the ball and “dishes out” passes in basketball, Jay has been handling drugs and passing them off in hand-to-hand sales since his teen years
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The “LQ” in this rhyme is famed Latin Quarter nightclub in NYC. From 1985-89, the club was a hip-hop Mecca, referenced in song by Boogie Down Productions, Ice-T, and others. See Schoolly D perform at the LQ in 1986 here