Question

How do solve `x^2+6x>=0` and write the answer as a inequality and interval notation?

Answer 1

The solutions are `x<=-6` and `x>=0`

or `x in] -oo,-6 ] uu [0, oo[`

Explanation:

Let `f(x)=x^2+6x`

Let's factorise the equation

`x^2+6x=x(x+6)`

The values when `f(x)=0` are `x=0` and ##x=-6

Let's do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaa)` `0` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+6` `color(white)(aaaaaa)` `-` `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<=-6` and `x>=0`

`x in] -oo,-6 ] uu [0, oo[`

graph{x^2+6x [-20.27, 20.27, -10.14, 10.14]}