Question

How do you solve `-3/(x+7)<=-4/(x+8)`?

Answer 1

The solution is `x in (-oo, -8)uu(-7, -4]`

Explanation:

We cannot do crossing over

`-3/(x+7)<=-4/(x+8)`

`-3/(x+7)+4/(x+8)<=0`

place everything on the same denominator

`(-3(x+8)+4(x+7))/((x+7)(x+8))<=0`

`(-3x-24+4x+28)/((x+7)(x+8))<=0`

`(x+4)/((x+7)(x+8))<=0`

Let `f(x)=(x+4)/((x+7)(x+8))`

Now build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-8` `color(white)(aaaaa)` `-7` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+8` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+7` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` ###color(white)(aa)#`-` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+4` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` ###color(white)(aa)#`-` `color(white)(aa)` ###color(white)(aaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+`

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

Therefore,

`f(x)<=0` when `x in (-oo, -8)uu(-7, -4]`

graph{-3/(x+7)+4/(x+8) [-39.84, 17.9, -6.8, 22.07]}