Question

How do you solve `1/(x+2)>=1/3` using a sign chart?

Answer 1

The answer is `x in ] -2,1 ]`

Explanation:

To simplify the expression, we cannot do crossing over

`1/(x+2)>=1/3`

`1/(x+2)-1/3>=0`

`(3-(x+2))/(3(x+2))>=0`

`(3-x-2)/(3(x+2))=(1-x)/(3(x+2))>=0`

Let `f(x)=(1-x)/(3(x+2))`

The domain of `f(x)` is `D_f(x)=RR-{-2}`

Now, we can do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaaaaaa)` `1` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aa)` `∥` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `1-x` `color(white)(aaaaa)` `+` `color(white)(aa)` `∥` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)>=0`, when `x in ] -2,1 ]`