Question

How can I solve this: |n| ≥ n

I know it's infinite solutions, but why?

And also what about when |n| ≤ n
Why is it only zero and infinite positive solutions?

Answer 1

`|n| \geq n`

For
`n\leq 0`

`|n| \geq n \Leftrightarrow -n\geq n \Rightarrow 2n\leq0 \Rightarrow n\leq0`

For
`n>0`

`|n| \geq n \Leftrightarrow n\geq n \Rightarrow 0\geq0\Rightarrow x\in\mathbb{R}`

`x>0 \vee x\leq 0 \Rightarrow \boxed{x\in\mathbb{R}}`

-------------------------------------------------------------

`|n| \leq n`

For
`n\leq0`

`|n|\leq n\Leftrightarrow -n\leq n \Rightarrow 2n\geq 0 \Rightarrow n\geq 0\\ n\geq 0 \wedge n\leq 0 \Rightarrow n=0`

For
`n>0`

`|n|\leq n \Leftrightarrow n\leq n \Rightarrow 0 \leq 0\Rightarrow n\in \mathbb{R}\\ n\in \mathbb{R} \wedge n>0 \Rightarrow n>0`

`n>0 \vee n=0 \Rightarrow \boxed{x\geq0}`