How do you solve $3/(x+1)<=3$ using a sign chart?

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How do you solve $3/(x+1)<=3$ using a sign chart?

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Answer 1 · 2017-06-22T14:06:06

The solution is $x in (-oo,-1) uu [0,+oo)$

Explanation:

We cannot do crossing over

Let's rearrange the inequality

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

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

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

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

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

Let's build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaa)$ $-1$ $color(white)(aaaaaaaa)$ $0$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $-3x$ $color(white)(aaaaaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aa)$ $0$ $color(white)(aa)$ $-$

$color(white)(aaaa)$ $x+1$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aa)$ $0$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$ $color(white)(aa)$ $0$ $color(white)(aa)$ $-$

Therefore,

$f(x)<=0$ when $x in (-oo,-1) uu [0,+oo)$ graph{3/(x+1)-3 [-19.3, 21.26, -12.45, 7.82]}

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How do you solve $3/(x+1)<=3$ using a sign chart?

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