# How do you solve x/(3-x)>2/(x+5) using a sign chart?

Feb 1, 2017

The answer is x in ]-7.77, -5[uu ]0.77,3[

#### Explanation:

We cannot do crossing over

We rewrite the equation

$\frac{x}{3 - x} > \frac{2}{x + 5}$

$\frac{x}{3 - x} - \frac{2}{x + 5} > 0$

$\frac{x \left(x + 5\right) - 2 \left(3 - x\right)}{\left(3 - x\right) \left(x + 5\right)} > 0$

$\frac{{x}^{2} + 5 x - 6 + 2 x}{\left(3 - x\right) \left(x + 5\right)} = \frac{{x}^{2} + 7 x - 6}{\left(3 - x\right) \left(x + 5\right)}$

Let $f \left(x\right) = \frac{{x}^{2} + 7 x - 6}{\left(3 - x\right) \left(x + 5\right)}$

The roots of ${x}^{2} + 7 x - 6$ are

$x = \frac{- 7 \pm \sqrt{49 + 4 \cdot 6}}{2}$

$x = \frac{- 7 \pm \sqrt{73}}{2}$

${x}_{1} = \frac{- 7 + 8.54}{2} = 0.77$

${x}_{2} = \frac{- 7 - 8.54}{2} = - 7.77$

Now, we can 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 a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 7.77$$\textcolor{w h i t e}{a a a a}$$- 5$$\textcolor{w h i t e}{a a a a}$$0.77$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 7.77$$\textcolor{w h i t e}{a a a a a}$$-$$\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}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$||color(white)(aaa)$+$

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

$\textcolor{w h i t e}{a a a a}$$x - 0.77$$\textcolor{w h i t e}{a a a a a a}$$-$$\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}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$||color(white)(aaa)$+$

$\textcolor{w h i t e}{a a a a}$$3 - x$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\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}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$||color(white)(aaa)$-$

$\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 a a a}$$-$$\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}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$||color(white)(aaa)$-$

Therefore,

$f \left(x\right) > 0$ when x in ]-7.77, -5[uu ]0.77,3[