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Solve this inequality

  • #1
utkarshakash
Gold Member
855
13

Homework Statement


[itex]x^2 \geq [x]^2[/itex]

[] denotes Greatest Integer Function
{} denotes Fractional Part

Homework Equations



The Attempt at a Solution



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

Answers and Replies

  • #2
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,312
1,001

Homework Statement


[itex]x^2 \geq [x]^2[/itex]

[] denotes Greatest Integer Function
{} denotes Fractional Part

Homework Equations



The Attempt at a Solution



[itex] x^2-[x]^2 \geq 0 \\
(x+[x])(x-[x]) \geq 0 [/itex]
How do you go from the above step to the next step.

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

[itex]-[x] \leq x \leq [x] \\
[/itex]
Considering left inequality
[itex]
x \geq -[x] \\
\left\{x\right\} \geq -2[x]
[/itex]
 
  • #3
utkarshakash
Gold Member
855
13
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

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

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