Question

How do you solve `\frac { 2} { x + 4} \geq \frac { 2} { x - 4}`?

Answer 1

There is no answer.

Refer to the process in the explanation.

Explanation:

Solve:

`2/(x+4)>=2/(x-4)`

Multiply both sides by `(x+4)`.

`(x+4)xx2/(x+4)>=(2(x+4))/(x-4)`

Cancel.

`color(red)cancel(color(black)((x+4)))xx2/color(red)cancel(color(black)(x+4))>=(2(x+4))/(x-4)`

Simplify.

`2>=(2(x+4))/(x-4)`

Multiply both sides by `(x-4)`.

`2xx(x-4)>=(2(x+4))/(x-4)xx(x-4)`

Cancel.

`2xx(x-4)>=(2(x+4))/color(red)cancel(color(black)(x-4))xxcolor(red)cancel(color(black)((x-4))`

Simplify.

`2(x-4)>=2(x+4)`

Divide each side by `2`.

`(2(x-4))/2>=(2(x+4))/2`

Cancel.

`(color(red)cancel(color(black)(2))(x-4))/color(red)cancel(color(black)(2))>=(color(red)cancel(color(black)(2))(x+4))/color(red)cancel(color(black)(2))`

Simplify.

`x-4>=x+4`

Subtract `x` from both sides.

`x-x-4>=x-x+4`

Cancel.

`color(red)cancel(color(black)(x))-color(red)cancel(color(black)(x))-4>=color(red)cancel(color(black)(x))-color(red)cancel(color(black)(x))+4`

Simplify.

`-4>=4`

This is impossible, so there is no answer.

Answer 2

The solution is `=x in (-4,4)`

Explanation:

Let's rearrange and simplify the inequality

We cannot do crossing over

`2/(x+4)>=2/(x-4)`

`2/(x+4)-2/(x-4)>=0`

`(2(x-4)-2(x+4))/((x+4)(x-4))`

`(2x-8-2x-8)/((x+4)(x-4))>=0`

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

Let `f(x)=-16/((x+4)(x-4))=16/((x+4)(4-x))`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-4` `color(white)(aaaaaaaaa)` `4` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+4` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `4-x` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-`

Therefore,

`f(x)>=0`, when `x in (-4,4)`

graph{2/(x+4)-2/(x-4) [-18.02, 18.01, -9.05, 8.97]}