# How do solve (x+4)/(1-x)<=0 and write the answer as a inequality and interval notation?

Feb 18, 2017

The answer is $- \infty < x \le - 4$ or $1 < x < + \infty$
x in ]-oo,-4]uu]1,+oo[ in interval notation

#### Explanation:

We solve this equation with a sign chart

Let $f \left(x\right) = \frac{x + 4}{1 - x}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{1\right\}$

Now, we 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}$$- 4$$\textcolor{w h i t e}{a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 4$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\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}$$1 - x$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$$\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}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) \le 0$ when x in ]-oo,-4]uu]1,+oo[ in interval notation

As an inequality $- \infty < x \le - 4$ or $1 < x < + \infty$