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How can an inequality with fractions should be solved ?

Let say :

$ \displaystyle \frac{2}{4}\quad?\quad\frac{5}{21} $

Please give me examples, information (step by step).

I should multiply over-cross ' to see if the equation is correct

--------------------------------------------------------- > Solved

Used: a/b = c/d => a*d = b*c

44 = 20 , becouse it not the same on both side, the fraction is wrong.

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    Thanks alot, b$u$t i solved it. :)2012-02-15

1 Answers 1

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The main idea is given in my comment: of course, you can use a cross-multiplication to solve this inequality - but why does it work? There is an rule (which is an axiom for inequalities) that if $a then for any positive $k$ it holds that $ka and for any negative $l$ it holds that $la>lb$.

Let us consider your example, you have $ \displaystyle{\frac 24 \quad?\quad \frac{5}{21}}. $ Whatever sign $?$ denotes, if we multiply both sides by a positive number, the sign does not change. So we multiply both sides by both denominators and obtain $ 21\times 4\times \frac24 \quad?\quad 4\times 21\times\frac{5}{21} $ and hence $ 42\quad?\quad20 $ so $?$ is $>$.

Then what about the cross-multiplication? You do the same but you write instead $ 21\times \left(4\times \frac24\right) \quad?\quad 4\times \left(21\times\frac{5}{21}\right) $ and since the denominators cancel it is equivalent to the cross-multiplication rule: $ 21\times 2\quad ?\quad 4\times 5. $

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    @user1022734: sorry, I didn't get you. Do you know how to simplify $24/4$?2012-02-17