How do you solve $x/(x+2)>=2$ ?

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Answer 1 · 2017-01-01T16:15:24

The answer is $x in [-4, 2[$

You cannot do crossing over

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

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

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

$(x-2x-4)/(x+2)>=0$

$(-x-4)/(x+2)>=0$

Let $f(x)=(-x-4)/(x+2)$

We can do a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaaaa)$ $-oo$ $color(white)(aaaaa)$ $-4$ $color(white)(aaaaaaa)$ $-2$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $-x-4$ $color(white)(aaaaa)$ $+$ $color(white)(aaaa)$ $0$ $color(white)(aaa)$ $-$ $color(white)(aa)$ $∥$ $color(white)(aa)$ $-$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaaaa)$ $-$ $color(white)(aaaa)$ #color(white)(aaa)-# $color(white)(aaa)$ $∥$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aaaa)$ #color(white)(aaa)+# $color(white)(aaa)$ $∥$ $color(white)(aa)$ $-$

Therefore,

$f(x)>=0$ , when $x in [-4, 2[$

How do you solve x/(x+2)>=2x/(x+2)>=2?