How do you solve $(2x+1) / (x-9) >= 0$ ?

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Answer 1 · 2017-03-01T15:54:38

The solution is $x in ]-oo, -1/2]uu]9,+oo[$

We solve this inequality with a sign chart

Let $f(x)=(2x+1)/(x-9)$

We can now build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-1/2$ $color(white)(aaaaaa)$ $9$ $color(white)(aaaaaaaa)$ $+oo$

$color(white)(aaaa)$ $2x+1$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-9$ $color(white)(aaaa)$ $-$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $+$ $color(white)(aaaaa)$ $-$ $color(white)(aaa)$ $||$ $color(white)(aaaa)$ $+$

Therefore,

$f(x)>=0$ when $x in ]-oo, -1/2]uu]9,+oo[$

How do you solve (2x+1) / (x-9) >= 0(2x+1) / (x-9) >= 0?