How do solve $(3-x)/(x+5)<=0$ and write the answer as a inequality and interval notation?

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Answer 1 · 2017-05-30T09:33:51

The solution is $x in (-oo,-5) uu [3, +oo)$ or
$x<-5$ and $x>=3$

Let $f(x)=(3-x)/(x+5)$

We build a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaa)$ $-5$ $color(white)(aaaaaaa)$ $3$ $color(white)(aaaaaa)$ $+oo$

$color(white)(aaaa)$ $x+5$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $3-x$ $color(white)(aaaaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $-$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $-$

Therefore,

$f(x)<=0$ when $x in (-oo,-5) uu [3, +oo)$ in interval notation or

$x<-5$ and $x>=3$ as inequality

How do solve (3-x)/(x+5)<=0(3-x)/(x+5)<=0 and write the answer as a inequality and interval notation?