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 $1 < x <+oo$

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