Question

How do you solve the inequality `-x^2-6x+7<=0`?

Answer 1

The answer is `x in ] -oo,-7] uu [ 1, +oo[`

Explanation:

Let`s rewrite the inequality

`x^2+6x-7>=0`

We factorise

`(x-1)(x+7)>=0`

Let `f(x)=(x-1)(x+7)`

The domain of `f(x)` is `D_f(x)=RR`

We do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaa)` `-7` `color(white)(aaaaa)` `1` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+7` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

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

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

Therefore,

`f(x)>=0`, when `x in ] -oo,-7] uu [ 1, +oo[`