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)xcolor(white)(aaaa)-oocolor(white)(aaaa)-1color(white)(aaaaaaa)0color(white)(aaaaaa)+oo
color(white)(aaaa)x+1color(white)(aaaa)-color(white)(aaaaaa)+color(white)(aa)∥color(white)(aa)+
color(white)(aaaa)xcolor(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 [