Question

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

Answer 1

`(-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]`