# How do you solve -x/(x-3)>=0?

Dec 9, 2016

$\left\{x \in R | 0 \le x \le 3\right\}$

#### Explanation:

Make a sign chart (y-chart) for the function

$y = \left(- 1\right) \frac{x}{x - 3}$

by making each factor a row and the zeros of each factor forming the columns across the top in number line order:

color(white)(aaaaaaaaaaaaaaaacolor(black)(0)aaaaaaaaacolor(black)(3)aaaaa

(-1)color(white)(aaaacolor(black)((-))aaaaaaacolor(black)((-))aaaaaaacolor(black)((-))
color(white)(aa)xcolor(white)(aa.aaacolor(black)((-))aaaaaaacolor(black)((+))aaaaaaacolor(black)((+))
(x-3)color(white)(aaacolor(black)((-))aaaaaaacolor(black)((-))aaaaaaacolor(black)((+))
$\frac{\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a}}{\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a}}$
color(white)(aaa)ycolor(white)(aaaaacolor(black)((-))aaaaaaacolor(black)((+))aaaaaaacolor(black)((-))

The sign of the y-value in the bottom row comes from the product of the factors above. $\left(-\right) \left(-\right) \left(-\right) = \left(-\right)$, $\left(-\right) \left(-\right) = \left(+\right)$, and so on.

From the chart it is clear that the function is positive in the interval from zero to 3, and since the inequality is "$\ge 0$," the endpoints are included. Thus the solution is the values of x where $0 \le x \le 3$.