How do you solve $(5x-8)/(x-5)>=2$ using a sign chart?

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Answer 1 · 2017-02-06T11:07:00

The answer is $x in ]-oo, -2/3 ] uu ] 5, +oo[$

Let's do some simplifications

$(5x-8)/(x-5)>=2$

$(5x-8)/(x-5)-2>=0$

$((5x-8)-2(x-5))/(x-5)>=0$

$(3x+2)/(x-5)>=0$

Let $f(x)=(3x+2)/(x-5)$

Let's build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaaa)$ $-oo$ $color(white)(aaaa)$ $-2/3$ $color(white)(aaaa)$ $5$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $3x+2$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-5$ $color(white)(aaaaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$

Therefore,

$f(x)>=0$ when $x in ]-oo, -2/3 ] uu ] 5, +oo[$

How do you solve (5x-8)/(x-5)>=2(5x-8)/(x-5)>=2 using a sign chart?