# How do you solve x-10/(x-1)>=4 using a sign chart?

Nov 18, 2016

The answer is x in ]-oo,-1 ]uu [6, +oo[

#### Explanation:

Let's do some simplification

$x - \frac{10}{x - 1} \ge 4$

Multiply by $\left(x - 1\right)$

$x \left(x - 1\right) - 10 \ge 4 \left(x - 1\right)$

${x}^{2} - x - 10 \ge 4 x - 4$

${x}^{2} - 5 x - 6 \ge 0$

Factorising

$\left(x + 1\right) \left(x - 6\right) \ge 0$

Let $f \left(x\right) = \left(x + 1\right) \left(x - 6\right)$

let's do a 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}$$- 1$$\textcolor{w h i t e}{a a a a}$$6$$\textcolor{w h i t e}{a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$\left(x - 6\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{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 a a}$$+$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$

$f \left(x\right) \ge 0$, when  x in ]-oo,-1 ]uu [6, +oo[

graph{(x+1)(x-6) [-17.02, 15.02, -12.81, 3.21]}