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

1 Answer
Feb 19, 2017

Answer:

The solution is #x in ]-3,2[#

Explanation:

We cannot do crossing over, so rewrite the inequality

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

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

Placing on the same denominator

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

#(3x+9-3x+6)/((x-2)(x+3))<=0#

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

Let #f(x)=15/((x-2)(x+3))#

The domain of #f(x)# is #D_f(x)=RR-{-3,2}#

Now, we can build the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaa)##-3##color(white)(aaaaaa)##2##color(white)(aaaaaaa)##+oo#

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

#color(white)(aaaa)##x-2##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aa)##-##color(white)(aa)##||##color(white)(aaaa)##+#

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

Therefore,

#f(x)<=0# when #x in ]-3,2[#