Question

How do you solve `-x ^ { 2} - 7x - 15\leq - 5`?

Answer 1

`x<=-5` or `x>=-2`

Explanation:

Writing
`-x^2-7x-15<=-5`

`-x^2-7x-10<=0`
`x^2+7x+10>=0`
so

`(x+7/2)^2-9/4>=0`
`(x+7/2-3/2)(x+7/2+3/2)>=0`
`(x+2)(x+5)>=0`
so
`x>=-2` or `x<=-5`

Answer 2

The solutions are `x in (-oo,-5] uu[-2,+oo)`

Explanation:

The inequality is

`-x^2-7x-15<=-5`

`x^2+7x-5+15>=0`

`x^2+7x+10>=0`

`(x+5)(x+2)>=0`

Let `f(x)=(x+5)(x+2)`

Build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-5` `color(white)(aaaaaa)` `-2` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+5` `color(white)(aaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in (-oo,-5] uu[-2,+oo)`

graph{x^2+7x+10 [-12.875, 7.125, -3.24, 6.76]}