Question

How do you solve `7-x^2<=0`?

Answer 1

The answer is `x in ]-oo,-sqrt7]uu[sqrt7,+oo[`

Explanation:

Let's factorise the inequality

`7-x^2<=0`

`(sqrt7+x)(sqrt7-x)<=0`

Let `f(x)=(sqrt7+x)(sqrt7-x)`

Now, we build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `-sqrt7` `color(white)(aaaa)` `sqrt7` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `sqrt7+x` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `sqrt7-x` `color(white)(aaaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `-`

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

Therefore,

`f(x)<=0` when `x in ]-oo,-sqrt7]uu[sqrt7,+oo[`