# How do you solve and write the following in interval notation: (x-2)/(x-3)<=-6?

Aug 4, 2017

The solution is $x \in \left[\frac{20}{7} , 3\right)$

#### Explanation:

We cannot do crossing over

Let's rearrange the inequality

$\frac{x - 2}{x - 3} \le - 6$

$\frac{x - 2}{x - 3} + 6 \le 0$

$\frac{\left(x - 2\right) + 6 \left(x - 3\right)}{x - 3} \le 0$

$\frac{\left(x - 2 + 6 x - 18\right)}{x - 3} \le 0$

$\frac{7 x - 20}{x - 3} \le 0$

Let $f \left(x\right) = \frac{7 x - 20}{x - 3}$

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 a a a}$$\frac{20}{7}$$\textcolor{w h i t e}{a a a a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$7 x - 20$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$0$$\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}$$x - 3$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

$\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 a}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\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,

$f \left(x\right) \le 0$ when $x \in \left[\frac{20}{7} , 3\right)$

graph{(x-2)/(x-3)+6 [-32.47, 32.47, -16.24, 16.25]}