How do you solve #-3/(x+7)<=-4/(x+8)#?

1 Answer
May 2, 2018

Answer:

The solution is #x in (-oo, -8)uu(-7, -4]#

Explanation:

We cannot do crossing over

#-3/(x+7)<=-4/(x+8)#

#-3/(x+7)+4/(x+8)<=0#

place everything on the same denominator

#(-3(x+8)+4(x+7))/((x+7)(x+8))<=0#

#(-3x-24+4x+28)/((x+7)(x+8))<=0#

#(x+4)/((x+7)(x+8))<=0#

Let #f(x)=(x+4)/((x+7)(x+8))#

Now build a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##-8##color(white)(aaaaa)##-7##color(white)(aaaa)##-4##color(white)(aaaa)##+oo#

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

#color(white)(aaaa)##x+7##color(white)(aaaaaa)##-##color(white)(aaaa)####color(white)(aa)##-##color(white)(aa)##||##color(white)(aa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##x+4##color(white)(aaaaaa)##-##color(white)(aaaa)####color(white)(aa)##-##color(white)(aa)####color(white)(aaa)##-##color(white)(aa)##0##color(white)(aa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##-##color(white)(aaa)##||##color(white)(aa)##+##color(white)(aa)##||##color(white)(aa)##-##color(white)(aa)##0##color(white)(aa)##+#

Therefore,

#f(x)<=0# when #x in (-oo, -8)uu(-7, -4]#

graph{-3/(x+7)+4/(x+8) [-39.84, 17.9, -6.8, 22.07]}