How to Solve an Inequality with Greatest Integer Function and Fractional Part?

[utkarshakash]

Homework Statement

$x^2 \geq [x]^2$

[] denotes Greatest Integer Function
{} denotes Fractional Part

Homework Equations

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
$x \geq -[x] \\ \left\{x\right\} \geq -2[x]$

 

[SammyS]

Homework Statement

$x^2 \geq [x]^2$

[] denotes Greatest Integer Function
{} denotes Fractional Part

Homework Equations

The Attempt at a Solution

$x^2-[x]^2 \geq 0 \\ (x+[x])(x-[x]) \geq 0$

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

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

$-[x] \leq x \leq [x] \\$
Considering left inequality
$x \geq -[x] \\ \left\{x\right\} \geq -2[x]$

 

[utkarshakash]

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

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

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

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