How do solve #(x+4)/(1-x)<=0# and write the answer as a inequality and interval notation?

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Answer 1 · 2017-02-18T07:42:00

The answer is #-oo < x <= -4# or # 1 < x <+oo #
#x in ]-oo,-4]uu]1,+oo[# in interval notation

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 #