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

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Answer 1 · 2016-12-21T13:22:37

The answer is #x in ] 2,4 [ #

We cannot do crossing over

#x/(x-2)-2>0#

#(x-2(x-2))/(x-2)>0#

#(4-x)/(x-2)>0#

Let #f(x)=(4-x)/(x-2)#

The domain of #f(x)# is #D_f(x)=RR-{2}#

Now we can do our sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##2##color(white)(aaaa)##4##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x-2##color(white)(aaaa)##-##color(white)(a)##∥##color(white)(a)##+##color(white)(aa)##+#

#color(white)(aaaa)##4-x##color(white)(aaaa)##+##color(white)(a)##∥##color(white)(a)##+##color(white)(aa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaa)##-##color(white)(a)##∥##color(white)(a)##+##color(white)(aa)##-#

Therefore,

#f(x)>0# when #x in ] 2,4 [ #