Question

How do you solve `12/(x+4)<=4`?

Answer 1

The solution is `x in (-oo,-4)uu [-1,+oo)`

Explanation:

You cannot do crossing over.

The inequality is

`(12)/(x+4)<=4`

`<=>`, `(12)/(x+4)-4<=0`

`<=>`, `(12-4(x+4))/(x+4)<=0`

`<=>`, `(12-16-4x)/(x+4)<=0`

`<=>`, `(-4-4x)/(x+4)<=0`

`<=>`, `(4(1+x))/(x+4)>=0`

Let `f(x)=(4(1+x))/(x+4)`

Build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+4` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aa)` #color(white)(aaa)-# `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aa)` `||` `color(white)(aa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in (-oo,-4)uu [-1,+oo)`

Answer 2

`12/(x+4)<=4` for `x<-4` and `x>=-1`

Explanation:

As the expression is undefined for `x=-4`, we want to stay away from that value.

Before we work on the expression algebraically, let's draw a graph:
graph{-(x+1)/(x+4) [-13.21, 6.79, -5.72, 4.28]}

Based on the graph we can see that the unequality is fulfilled for
`x<-4` and `x>=-1`

Let us clean up the expression to make it easier to work with:
An equivalent expression is

`3/(x+4)<=1`

`3/(x+4)-1<=0`

`(3-(x+4))/(x+4)<=0`

`-(x+1)/(x+4)<=0`

As this is undefined for `x=-4`, we need to consider two situations: `x> -4` and `x<-4`

1) `x> -4`: As `x+4>0` we can multiply both sides with the denominator `x+4` and still keep the sign of inequality:

`-((x+1)(x+4))/(x+4)<=0`

`-(x+1)<=0`

`x+1>=0`

`x>=-1`
Therefore `12/(x+4)<=4` when `x>=-1`

2) `x< -4`: Now the denominator `(x+4)` is negative, so if we multiply the unequality with the value of denominator, we have to turn the unequal sign around:

`-((x+1)(x+4))/(x+4)>=0`

`-(x+1)>=0`

`x+1<=0`

`x<=-1`

As the starting point was that `x<-4`, this means that `12/(x+4)<=4` for all `x<-4`