Can someone please help me with this absolute value inequality, can I draw a graph for it? #|(x+4)/(x+2)|<=1#?

1 Answer
Mar 27, 2018

#(-oo,-3]#

Explanation:

#|(x+4)/(x+2)|<=1#

We know by the absolute value property that we have to solve both:

#(x+4)/(x+2)<=1# and #-((x+4)/(x+2))<=1#

For:

#(x+4)/(x+2)<=1#

Subtract #1#

#(x+4)/(x+2)-1<=0#

Add LHS:

#((x+4)-(x+2))/(x+2)<=0#

Simplify:

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

Divide by 2:

#1/(x+2)<=0#

There is no solution for zero, (undefined division by zero)

only #x<-2#

For:

#-((x+4)/(x+2))<=1#

Subtract 1:

#-(x+4)/(x+2)-1<+0#

Multiply by #-1#:

#(x+4)/(x+2)+1<=0#

Add LHS:

#(2x+6)/(x+2)<=0#

Solving for zero:

#(2x+6)/(x+2)=0#

#x=-3#

We now need to look at :

#|(x+4)/(x+2)|-1<=0#

For:

#x<-3#

#0<=|(x+4)/(x+2)|<1#

So:

#|(x+4)/(x+2)|-1<=0#

For #x> -3#, #x!=-2#

#|(x+4)/(x+2)|>1#

So:

#|(x+4)/(x+2)|-1>0#

So only: #x<=-3#

Solution set:

#(-oo,-3]#