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 \\<br /> (x+[x])(x-[x]) \geq 0 \\<br /> -[x] \leq x \leq [x] \\<br />
Considering left inequality
<br /> x \geq -[x] \\<br /> \left\{x\right\} \geq -2[x]<br />

 

[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 \\<br /> (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] \\<br />
Considering left inequality
<br /> x \geq -[x] \\<br /> \left\{x\right\} \geq -2[x]<br />

 

[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)