当x<=-3时,|x+3|+|x-4|+|x+1|=-3x,有最小值9
当-3<=x<=-1时,|x+3|+|x-4|+|x+1|=-x+6,有最小值7
当-1<=x<=4时,|x+3|+|x-4|+|x+1|=x+8,有最小值7
当x>=4时,|x+3|+|x-4|+|x+1|=3x,有最小值12
综上所述,最小值是7,此时x=-1