Question

How do you solve and write the following in interval notation: `x^2 + 6x + 5 >= 0`?

Answer 1

The interval notation is `] -oo,-5 ] uu [-1,+oo[`

Explanation:

Let's factorise the equation
`y=x^2+6x+5>=0` `=>` `(x+1)(x+5)>=0`
Let's do a sign chart
`color(white)(aaaa)` `x` `color(white)(aaa)` `-oo` `color(white)(aaaa)` `-5` `color(white)(aaa)` `-1` `color(white)(aaa)` ##+oo #color(white)(aa)#`x+5` `color(white)(aaaaaaa)` `-` `color(white)(aaa)` `+` `color(white)(aaa)` `+` `color(white)(aaa)`
`color(white)(aa)` `x+1` `color(white)(aaaaaaa)` `-` `color(white)(aaa)` `-` `color(white)(aaa)` `+` `color(white)(aaa)`
`color(white)(aaaaa)` `y` `color(white)(aaaaaaa)` `+` `color(white)(aaa)` `-` `color(white)(aaa)` `+` `color(white)(aaa)`

So `y>=0` on the intervals `] -oo,-5 ] uu [-1,+oo[`
graph{x^2+6x+5 [-10.8, 5.004, -5.31, 2.59]}