Question

How do you solve `3x^2-11x-4<=0` using a sign chart?

Answer 1

The answer is `x in [-1/3,4]`

Explanation:

Let's factorise the expression

`3x^2-11x-4=(3x+1)(x-4)`

Let `f(x)=3x^2-11x-4`

Now, we can make the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaa)` `-1/3` `color(white)(aaaaa)` `4` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `3x+1` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<=0` when `x in [-1/3,4]`