Question

How do you solve `(x-2)/(2x+5)<0` using a sign chart?

Answer 1

The answer is `x in ] -5/2, 2[`

Explanation:

Let `f(x)=(x-2)/(2x+5)`

The domain of `f(x)` is `D_f(x)=RR-{-5/2}`

Now, we can do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-5/2` `color(white)(aaaaa)` `2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `2x+5` `color(white)(aaaa)` `-` `color(white)(aaa)` `∣∣` `color(white)(aa)` `+` `color(white)(aaa)` `+`

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

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

Therefore,

`f(x)<0`, when `x in ] -5/2, 2[`