# How do you solve (-x+8)/(x-2)>=5 using a sign chart?

Aug 22, 2017

Solution: $2 < x \le 3 \mathmr{and} \left(2 , 3\right]$

#### Explanation:

$\frac{- x + 8}{x - 2} \ge 5 \mathmr{and} \frac{- x + 8}{x - 2} - 5 \ge 0$ or

$\frac{- x + 8 - 5 x + 10}{x - 2} \ge 0 \mathmr{and} \frac{- 6 x + 18}{x - 2} \ge 0$ or

$\frac{- 6 \left(x - 3\right)}{x - 2} \ge 0 \mathmr{and} \frac{x - 3}{x - 2} \le 0$ Critical points are

$x = 2 , x = 3$ When x =3 ; (x-3)/(x-2) =0 ; x !=2

Sign chart:

when $x < 2$ ; sign of $\frac{x - 3}{x - 2}$ is (-)/(-)=(+) ; >0

when $2 < x < 3$ ; sign of $\frac{x - 3}{x - 2}$ is (-)/(+)=(-) ; < 0

when $x > 3$ ; sign of $\frac{x - 3}{x - 2}$ is (+)/(+)=(+) ; >0

Solution: $2 < x \le 3 \mathmr{and} \left(2 , 3\right]$ [Ans]