How do you solve #(2x)/(x-2)<=3# using a sign chart?

1 Answer
Jun 14, 2017

Answer:

The solution is #x in (-oo,2)uu [6,+oo)#

Explanation:

We cannot do crossing over

Let's rearrange the inequality

#(2x)/(x-2)<=3#

#(2x)/(x-2)-3<=0#

#(2x-3(x-2))/(x-2)<=0#

#(2x-3x+6)/(x-2)<=0#

#(6-x)/(x-2)<=0#

Let #f(x)=(6-x)/(x-2)#

Let's build the sign chart

#color(white)(aaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaaaa)##2##color(white)(aaaaaaa)##6##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##x-2##color(white)(aaaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(a)##0##color(white)(aaaa)##+#

#color(white)(aaaa)##6-x##color(white)(aaaaa)##+##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(a)##0##color(white)(aaaa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(a)##0##color(white)(aaaa)##-#

Therefore,

#f(x)<=0# when #x in (-oo,2)uu [6,+oo)#