# How do you solve (x+8)^2(x+5)(x+7)^2>=0 using a sign chart?

Jul 23, 2018

The solution is $x \in \left\{- 8\right\} \cup \left\{- 7\right\} \cup \left[- 5 , + \infty\right)$

#### Explanation:

The inequality is

${\left(x + 8\right)}^{2} \left(x + 5\right) {\left(x + 7\right)}^{2} \ge 0$

Let $f \left(x\right) = {\left(x + 8\right)}^{2} \left(x + 5\right) {\left(x + 7\right)}^{2}$

Let 's build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$- 8$$\textcolor{w h i t e}{a a a a a a}$$- 7$$\textcolor{w h i t e}{a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$${\left(x + 8\right)}^{2}$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$${\left(x + 7\right)}^{2}$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$color(white)(aaaa)+$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x + 5$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$color(white)(aaaa)-$\textcolor{w h i t e}{a a}$color(white)(aaaa)-$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a}$$0$$\textcolor{w h i t e}{a a}$$+$

Therefore,

$f \left(x\right) \ge 0$ when $x \in \left\{- 8\right\} \cup \left\{- 7\right\} \cup \left[- 5 , + \infty\right)$

graph{(x+8)^2(x+5)(x+7)^2 [-9.692, -3.533, -1.228, 1.85]}