We solve this equation with a sign chart
Let #f(x)=(x+4)/(1-x)#
The domain of #f(x)# is #D_f(x)=RR-{1}#
Now, we build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaa)##-4##color(white)(aaaaaa)##1##color(white)(aaaaaaa)##+oo#
#color(white)(aaaa)##x+4##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##+#
#color(white)(aaaa)##1-x##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##-#
#color(white)(aaaa)##f(x)##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##-#
Therefore,
#f(x)<=0# when #x in ]-oo,-4]uu]1,+oo[# in interval notation
As an inequality #-oo < x <= -4# or # 1 < x <+oo #