How do you solve $6/(x+1)<-3$ ?

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Answer 1 · 2017-05-24T15:23:12

The answer is $=x in (-3,-1)$

We cannot do crossing over.

Let's simplify the inequality

$6/(x+1)<-3$

$6/(x+1)+3<0$

$(6+3(x+1))/(x+1)<0$

$(6+3x+3)/(x+1)<0$

$(3x+9)/(x+1)<0$

$(3(x+3))/(x+1)<0$

Let $f(x)=(3(x+3))/(x+1)$

We can build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-3$ $color(white)(aaaaaaa)$ $-1$ $color(white)(aaaaaaa)$ $+oo$

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

$color(white)(aaaa)$ $x+1$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$

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

Therefore,

$f(x)<0$ when $x in (-3,-1)$

How do you solve 6/(x+1)<-36/(x+1)<-3?