# How do you solve 1/(x+2)>=1/3 using a sign chart?

Dec 26, 2016

The answer is x in ] -2,1 ]

#### Explanation:

To simplify the expression, we cannot do crossing over

$\frac{1}{x + 2} \ge \frac{1}{3}$

$\frac{1}{x + 2} - \frac{1}{3} \ge 0$

$\frac{3 - \left(x + 2\right)}{3 \left(x + 2\right)} \ge 0$

$\frac{3 - x - 2}{3 \left(x + 2\right)} = \frac{1 - x}{3 \left(x + 2\right)} \ge 0$

Let $f \left(x\right) = \frac{1 - x}{3 \left(x + 2\right)}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{- 2\right\}$

Now, we can do 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}$$- 2$$\textcolor{w h i t e}{a a a a a a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a}$∥$\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 a}$$1 - x$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a}$∥$\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 a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a}$∥$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) \ge 0$, when x in ] -2,1 ]