Question

How do you solve `5x<=2-3x^2` using a sign chart?

Answer 1

The solution is `x in [-2,1/3]`

Explanation:

Let's rearrange and factorise the inequality

`5x<=2-3x^2`

`3x^2+5x-2<=0`

`(3x-1)(x+2)<=0`

Let `f(x)=(3x-1)(x+2)`

We build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaa)` `1/3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

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

`color(white)(aaaa)` `f(x9` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<=0` when `x in [-2,1/3]`