How do you solve $1/(x+2)>=1/3$ using a sign chart?

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Answer 1 · 2016-12-26T15:52:52

The answer is $x in ] -2,1 ]$

To simplify the expression, we cannot do crossing over

$1/(x+2)>=1/3$

$1/(x+2)-1/3>=0$

$(3-(x+2))/(3(x+2))>=0$

$(3-x-2)/(3(x+2))=(1-x)/(3(x+2))>=0$

Let $f(x)=(1-x)/(3(x+2))$

The domain of $f(x)$ is $D_f(x)=RR-{-2}$

Now, we can do the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-2$ $color(white)(aaaaaaaaa)$ $1$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaa)$ $-$ $color(white)(aa)$ $∥$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

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

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

Therefore,

$f(x)>=0$ , when $x in ] -2,1 ]$

How do you solve 1/(x+2)>=1/31/(x+2)>=1/3 using a sign chart?