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

Nov 7, 2016

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

#### Explanation:

Let's factorise the equation
$y = {x}^{2} + 6 x + 5 \ge 0$ $\implies$$\left(x + 1\right) \left(x + 5\right) \ge 0$
Let's do a sign chart
$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 5$$\textcolor{w h i t e}{a a a}$$- 1$$\textcolor{w h i t e}{a a a}$+oo color(white)(aa)x+5$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$
$\textcolor{w h i t e}{a a}$$x + 1$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$
$\textcolor{w h i t e}{a a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$

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