Question

How do you solve `(x-4)/(x^2+2x)<=0` using a sign chart?

Answer 1

The answer is `x in ] -oo,-2 [ uu ] 0,4]`

Explanation:

The denominator is

`x^2+2x=x(x+2)`

Let `f(x)=(x-4)/(x(x+2))`

The domain of `f(x)` is `D_f(x)=RR-{0,-2}`

Now, we can do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `0` `color(white)(aaaa)` `4` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-` `color(white)(aa)` `∥` `color(white)()` `-` `color(white)()` `∥` `color(white)(a)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaa)` `-` `color(white)(aa)` `∥` `color(white)()` `-` `color(white)()` `∥` `color(white)(a)` `-` `color(white)(aaa)` `+`

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

So,
`f(x)<=0`, when `x in ] -oo,-2 [ uu ] 0,4]`