Question

How do you solve `((3x-2)(x-4))/(x+4)^2<0` using a sign chart?

Answer 1

The answer is `=x in ] 2/3 , 4 [`

Explanation:

Let `f(x)=((3x-2)(x-4))/(x+4)^2`

The domain of `f(x)` is `D_f(x)=RR-{-4}`

The denominator is `>0, AA x in D_f(x)`

Let's do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `2/3` `color(white)(aaaaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `3x-2` `color(white)(aaaa)` `-` `color(white)(aa)` `color(red)(∥)` `color(white)(a)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `3x-2` `color(white)(aaaa)` `-` `color(white)(aa)` `color(red)(∥)` `color(white)(a)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aa)` `color(red)(∥)` `color(white)(a)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<0`, when `x in ] 2/3 , 4 [`