How do you solve #x/(x+20)>2/(x+8)# using a sign chart?

1 Answer
May 29, 2017

Answer:

The solution is #x in (-oo,-20) uu (-10, -8) uu (4+oo)#

Explanation:

Let's rearrange the equation, we cannot do crossing over

#x/(x+20)>2/(x+8)#

#x/(x+20)-2/(x+8)>0#

The LCD is #(x+20)(x+8)#

So,

#(x(x+8)-2(x+20))/((x+20)(x+8))>0#

#(x^2+8x-2x-40)/((x+20)(x+8))>0#

#(x^2+6x-40)/((x+20)(x+8))>0#

#((x+10)(x-4))/((x+20)(x+8))>0#

Let #f(x)=((x+10)(x-4))/((x+20)(x+8))#

We can build the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-20##color(white)(aaaa)##-10##color(white)(aaaa)##-8##color(white)(aaaaa)##4##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+20##color(white)(aaaaa)##-##color(white)(aa)##||##color(white)(aaa)##+##color(white)(aaaa)##+##color(white)(aa)##||##color(white)(aa)##+##color(white)(aa)##+#

#color(white)(aaaa)##x+10##color(white)(aaaaa)##-##color(white)(aa)##||##color(white)(aaa)##-##color(white)(aaaa)##+##color(white)(aa)##||##color(white)(aa)##+##color(white)(aa)##+#

#color(white)(aaaa)##x+8##color(white)(aaaaaa)##-##color(white)(aa)##||##color(white)(aaa)##-##color(white)(aaaa)##-##color(white)(aa)##||##color(white)(aa)##+##color(white)(aa)##+#

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

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

Therefore,

#f(x)>0# when #x in (-oo,-20) uu (-10, -8) uu (4+oo)#