Question

How do you solve `9x^2+3x-2>=0` using a sign chart?

Answer 1

The answer is `x in ] -oo,-2/3 ] uu [1/3, +oo[`

Explanation:

Let's factorise the LHS

`9x^2+3x-2=(3x-1)(3x+2)`

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

Now we can make the sign chart

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

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

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

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

Therefore,

`f(x)>=0`, when `x in ] -oo,-2/3 ] uu [1/3, +oo[`