Question

How do you solve `x/(x+20)>2/(x+8)` using a sign chart?

Answer 1

The solution is `x in (-oo,-20) uu (-10, -8) uu (4+oo)`

Explanation:

Let's rearrange the equation, we cannot do crossing over

`x/(x+20)>2/(x+8)`

`x/(x+20)-2/(x+8)>0`

The LCD is `(x+20)(x+8)`

So,

`(x(x+8)-2(x+20))/((x+20)(x+8))>0`

`(x^2+8x-2x-40)/((x+20)(x+8))>0`

`(x^2+6x-40)/((x+20)(x+8))>0`

`((x+10)(x-4))/((x+20)(x+8))>0`

Let `f(x)=((x+10)(x-4))/((x+20)(x+8))`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-20` `color(white)(aaaa)` `-10` `color(white)(aaaa)` `-8` `color(white)(aaaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+20` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `+` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aa)` `+`

`color(white)(aaaa)` `x+10` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aaa)` `-` `color(white)(aaaa)` `+` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aa)` `+`

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

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

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

Therefore,

`f(x)>0` when `x in (-oo,-20) uu (-10, -8) uu (4+oo)`