How do you solve #x/(x-3)<=-8/(x-6)# using a sign chart?

1 Answer
Mar 15, 2017

Answer:

The solution is # x in [-6,3[ uu [4,6 [#

Explanation:

We cannot do crossing over

#x/(x-3)<=-8/(x-6)#

We rearrange and factorise the inequality

#x/(x-3)+8/(x-6)<=0#

#(x(x-6)+8(x-3))/((x-3)(x-6))<=0#

#(x^2-6x+8x-24)/((x-3)(x-6))<=0#

#(x^2+2x-24)/((x-3)(x-6))<=0#

#((x-4)(x+6))/((x-3)(x-6))<=0#

Let #f(x)=((x-4)(x+6))/((x-3)(x-6))#

We can build the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-6##color(white)(aaaaaaa)##3##color(white)(aaaaa)##4##color(white)(aaaaaaa)##6##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+6##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aa)##+##color(white)(aa)##+##color(white)(aaaa)##||##color(white)(aa)##+#

#color(white)(aaaa)##x-3##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+##color(white)(aa)##+##color(white)(aaaa)##||##color(white)(aa)##+#

#color(white)(aaaa)##x-4##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##-##color(white)(aa)##+##color(white)(aaaa)##||##color(white)(aa)##+#

#color(white)(aaaa)##x-6##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##-##color(white)(aa)##-##color(white)(aaaa)##||##color(white)(aa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##+##color(white)(aa)##-##color(white)(aaaa)##||##color(white)(aa)##+#

Therefore,

#f(x)<=0# when # x in [-6,3[ uu [4,6 [#