# How do you solve 1/4<7/(7-x) using a sign chart?

##### 1 Answer
Jun 6, 2017

The solution is $x \in \left(- 21 , 7\right)$

#### Explanation:

We cannot do crossing over.

Let's rearrange the equation

$\frac{1}{4} < \frac{7}{7 - x}$

$\frac{7}{7 - x} - \frac{1}{4} > 0$

$\frac{28 - \left(7 - x\right)}{4 \left(7 - x\right)} > 0$

$\frac{21 + x}{4 \left(7 - x\right)} > 0$

Let $f \left(x\right) = \frac{21 + x}{4 \left(7 - x\right)}$

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}$$- 21$$\textcolor{w h i t e}{a a a a a a a}$$7$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$21 + x$$\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 a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

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

Therefore,

$f \left(x\right) > 0$ when $x \in \left(- 21 , 7\right)$