Here's one way:
library(ggplot2)
library(reshape2)
nwpt <- subset(nonwork,
select=c(DID,Distance,PersonTrips1995,PersonTrips2001,PersonTrips2009))
nwpt <- melt(nwpt, id.vars=c("DID","Distance"))
ggplot(data=nwpt, aes(x=DID, y=value, colour=variable)) +
geom_line(size=0.5) +
labs(title="Non Work Person Trips", y="Percent") +
scale_x_discrete(name="Distance", labels=nwpt$Distance) +
theme(axis.text.x=element_text(angle=90))
Produces this with your dataset: