How do you solve $6/x<2$ using a sign chart?

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Answer 1 · 2017-06-11T16:18:03

The solution is $x in (-oo,0) uu (3,+oo)$

Let's rewrite and simplify the inequality

We cannot do crossing over

$6/x<2$

$6/x-2<0$

$(6-2x)/x<0$

$(2(3-x))/x<0$

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

Let's build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaa)$ $0$ $color(white)(aaaaaaa)$ $3$ $color(white)(aaaaa)$ $+oo$

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

$color(white)(aaaa)$ $(3-x)$ $color(white)(aaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaa)$ $+$ $color(white)(aaaa)$ $-$

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

Therefore,

$f(x)<0$ , when $x in (-oo,0) uu (3,+oo)$ graph{(6/x)-2 [-22.81, 22.8, -11.4, 11.42]}

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