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September 24th, 2017, 12:05 PM   #1
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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?
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September 24th, 2017, 12:30 PM   #2
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you can square inequalities but they don't behave as you have written.

let $b^2 < a^2$

then $-a < b < a$
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September 24th, 2017, 01:20 PM   #3
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Any positive number is > any negative number. However when you square them the absolute values of the original numbers determine the relationship.
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September 24th, 2017, 01:31 PM   #4
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Quote:
Originally Posted by Antoniomathgini View Post
Am I doing something wrong here, or is it illegal to square both sides of an inequality all together?
You have multiplied the left hand side of your inequality by (5) and the right hand side by (-5). Does that give you a hint as to the problem?

Quote:
Originally Posted by romsek View Post
you can square inequalities but they don't behave as you have written.
I guess if both sides are positive, you can. You could also work out a proper treatment for when both sides are negative. Come to that, there is a proper treatment for the sides having different sign, but you are mostly working on a case-by-case basis.

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 mixed-sign 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.
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Last edited by v8archie; September 24th, 2017 at 01:37 PM.
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