How do you solve $(3x+6)/(2x-12)<=0$ using a sign chart?

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How do you solve $(3x+6)/(2x-12)<=0$ using a sign chart?

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Answer 1 · 2017-11-03T14:31:32

The solution is $x in [-2,6)$

Explanation:

Let $f(x)=(3x+6)/(2x-12)=(3(x+2))/(2(x-6))$

Let's build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaa)$ $-2$ $color(white)(aaaaaaa)$ $6$ $color(white)(aaaaaa)$ $+oo$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $0$ $color(white)(aaa)$ $+$ $color(white)(aaaaaa)$ $+$

$color(white)(aaaa)$ $x-6$ $color(white)(aaaaaa)$ $-$ $color(white)(aaa)$ $0$ $color(white)(aaa)$ $-$ $color(white)(aa)$ $||$ $color(white)(aaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaa)$ $+$ $color(white)(aaa)$ $0$ $color(white)(aaa)$ $-$ $color(white)(aa)$ $||$ $color(white)(aaa)$ $+$

Therefore,

$f(x)<=0$ when $x in [-2,6)$

graph{(3x+6)/(2x-12) [-22.8, 22.83, -11.4, 11.4]}

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How do you solve $(3x+6)/(2x-12)<=0$ using a sign chart?

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