How do you solve #p/(p-16)+2/(p-6)<=0# using a sign chart?

1 Answer
May 7, 2017

Answer:

The solution is #p in [-4,6) uu [8, 16)#

Explanation:

Let 's simplify the #LHS# of the inequality

#p/(p-16)+2/(p-6)<=0#

#(p(p-6)+2(p-16))/((p-16)(p-6))<=0#

#(p^2-6p+2p-32)/((p-16)(p-6))<=0#

#(p^2-4p-32)/((p-16)(p-6))<=0#

#((p+4)(p-8))/((p-16)(p-6))<=0#

Let #f(p)=((p+4)(p-8))/((p-16)(p-6))#

We can build the sign chart

#color(white)(aaaa)##p##color(white)(aaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaaaaa)##6##color(white)(aaaaaa)##8##color(white)(aaaaaaa)##16##color(white)(aaaa)##+oo#

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

#color(white)(aaaa)##p-6##color(white)(aaaa)##-##color(white)(aaaaaa)##-##color(white)(aaa)##||##color(white)(aaa)##+##color(white)(aaa)##+##color(white)(aaa)##||##color(white)(aa)##+#

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

#color(white)(aaaa)##p-16##color(white)(aaa)##-##color(white)(aaaaaa)##-##color(white)(aaa)##||##color(white)(aaa)##-##color(white)(aaa)##-##color(white)(aaa)##||##color(white)(aa)##+#

#color(white)(aaaa)##f(p)##color(white)(aaaaa)##+##color(white)(aaaaaa)##-##color(white)(aaa)##||##color(white)(aaa)##+##color(white)(aaa)##-##color(white)(aaa)##||##color(white)(aa)##+#

Therefore,

#f(p)<=0# when #p in [-4,6) uu [8, 16)#