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

1 Answer
Dec 9, 2016

Answer:

#{x in R | 0 <= x <= 3}#

Explanation:

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

#y=(-1)(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)((+))#
#color(white)(aaaaaaaaaaaaaaaaaaaaaa)/color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa)#
#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. #( - ) (- )( - ) = (- ) #, # (-)(-) = (+)#, 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 "#>= 0#," the endpoints are included. Thus the solution is the values of x where #0<=x<=3#.