How do you solve #1/(x+2)>=1/3# using a sign chart?

1 Answer
Dec 26, 2016

Answer:

The answer is #x in ] -2,1 ] #

Explanation:

To simplify the expression, we cannot do crossing over

#1/(x+2)>=1/3#

#1/(x+2)-1/3>=0#

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

#(3-x-2)/(3(x+2))=(1-x)/(3(x+2))>=0#

Let #f(x)=(1-x)/(3(x+2))#

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

Now, we can do the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaaaaaaa)##1##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+2##color(white)(aaaaa)##-##color(white)(aa)##∥##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##1-x##color(white)(aaaaa)##+##color(white)(aa)##∥##color(white)(aaaa)##+##color(white)(aaaa)##-#

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

Therefore,

#f(x)>=0#, when #x in ] -2,1 ] #