Question

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

Answer 1

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

Explanation:

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]}