# Solve this inequality

1. Apr 22, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
$x^2 \geq [x]^2$

[] denotes Greatest Integer Function
{} denotes Fractional Part
2. Relevant equations

3. The attempt at a solution

$x^2-[x]^2 \geq 0 \\ (x+[x])(x-[x]) \geq 0 \\ -[x] \leq x \leq [x] \\$
Considering left inequality
$x \geq -[x] \\ \left\{x\right\} \geq -2[x]$

2. Apr 22, 2013

### SammyS

Staff Emeritus
How do you go from the above step to the next step.

(It does look valid, but an explanation seems to be in order.)

3. Apr 23, 2013

### utkarshakash

Ah! I made a silly mistake there. Actually it should be like this

$x \in \left( -∞, -[x] \right] U \left[ [x],∞ \right)$