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

Jan 2, 2017

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

#### Explanation:

Let`s rewrite the inequality

${x}^{2} + 6 x - 7 \ge 0$

We factorise

$\left(x - 1\right) \left(x + 7\right) \ge 0$

Let $f \left(x\right) = \left(x - 1\right) \left(x + 7\right)$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R}$

We do a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$- 7$$\textcolor{w h i t e}{a a a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$x - 1$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a 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 a}$$-$$\textcolor{w h i t e}{a a a a a}$$+$

Therefore,

$f \left(x\right) \ge 0$, when  x in ] -oo,-7] uu [ 1, +oo[