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September 24th, 2017, 12:05 PM  #1 
Member Joined: Apr 2017 From: Canada Posts: 32 Thanks: 2  Is the square operation in inequalities?
$\displaystyle 5 > 5$ $\displaystyle 5^2 > (5)^2$ Does one reverse the signs at this point? (Since one of the sides is multiplied by a negative?) Either way, you arrive at the contradiction: $\displaystyle 25>25$ Am I doing something wrong here, or is it illegal to square both sides of an inequality all together? 
September 24th, 2017, 12:30 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,780 Thanks: 919 
you can square inequalities but they don't behave as you have written. let $b^2 < a^2$ then $a < b < a$ 
September 24th, 2017, 01:20 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,443 Thanks: 564 
Any positive number is > any negative number. However when you square them the absolute values of the original numbers determine the relationship.

September 24th, 2017, 01:31 PM  #4  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,214 Thanks: 2410 Math Focus: Mainly analysis and algebra  Quote:
Quote:
The trick works only if both sides are positive because you multiply the larger side by a larger number. That is: $$5 \gt 3 \implies 25 = 5\times5 \gt 3\times5 \gt 3\times 3 = 9$$ Similar thinking goes into the negative and mixedsign cases. I would suggest that the best policy is thinking carefully about what you are doing to what quantities. This is especially true when you have variables that might be positive or negative. Last edited by v8archie; September 24th, 2017 at 01:37 PM.  

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