Question

How do you solve the inequality `5+1/x>16/x`?

Answer 1

The answer is `x in ] -oo,0 [ uu ] 3, +oo[`

Explanation:

Let's rewrite the inequality

`5+1/x-16/x>0`

`5-15/x>0`

`(5(x-3))/x>0`

Let `f(x)=(5(x-3))/x`

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

Now we can do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `0` `color(white)(aaaaaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>0`, when `x in ] -oo,0 [ uu ] 3, +oo[`