We cannot cross over.
Let's rewrite the inequality
#(x+3)/x+2<=0#
#(x+3+2x)/(x)<=0#
#(3x+3)/(x)<=0#
#(3(x+1))/x<=0#
Let #f(x)=(3(x+1))/x#
Let's do the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaaaaa)##0##color(white)(aaaaaa)##+oo#
#color(white)(aaaa)##x+1##color(white)(aaaa)##-##color(white)(aaaaaa)##+##color(white)(aa)##∥##color(white)(aa)##+#
#color(white)(aaaa)##x##color(white)(aaaaaaaa)##-##color(white)(aaaaaa)##-##color(white)(aa)##∥##color(white)(aa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aaaaa)##-##color(white)(aa)##∥##color(white)(aa)##+#
Therefore,
#f(x)<=0#, when #x in [-1, 0 [ #