How do you solve $x/(x-2)>2$ using a sign chart?

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Answer 1 · 2016-12-21T13:22:37

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

We cannot do crossing over

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

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

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

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

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

Now we can do our sign chart

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

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

$color(white)(aaaa)$ $4-x$ $color(white)(aaaa)$ $+$ $color(white)(a)$ $∥$ $color(white)(a)$ $+$ $color(white)(aa)$ $-$

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

Therefore,

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

How do you solve x/(x-2)>2x/(x-2)>2 using a sign chart?