Question

How do you solve `6x^2+3x>= -5x-3+x^2`?

Answer 1

The answer is `x in ]-oo,-1]uu [-3/5, +oo[`

Explanation:

Let'e rearrange the inequality

`6x^2+3x>=-5x-3+x^2`

`5x^2+8x+3>=0`

We factorise

`(5x+3)(x+1)>=0`

Let `f(x)=(5x+3)(x+1)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `-3/5` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `5x+3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x>=0)` when `x in ]-oo,-1]uu [-3/5, +oo[`