Multiplying the two inequalities

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SUMMARY

The discussion centers on the multiplication of two inequalities: x - y ≤ a - b ≤ x + y and t - g ≤ c - d ≤ t + g. It is established that term-by-term multiplication is valid only if all terms are positive. If the signs of the terms are uncertain, caution is advised as valid multiplication cannot be guaranteed. When all terms are confirmed positive, the inequalities can be rearranged before multiplication to ensure accuracy.

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  • Understanding of basic algebraic inequalities
  • Familiarity with the properties of positive numbers
  • Knowledge of rearranging inequalities
  • Experience with mathematical proofs and logical reasoning
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  • Study the properties of inequalities in algebra
  • Learn about the conditions for multiplying inequalities
  • Explore examples of rearranging inequalities for multiplication
  • Investigate the implications of negative numbers in inequality operations
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Mathematics students, educators, and anyone involved in algebraic problem-solving who seeks to deepen their understanding of inequalities and their manipulation.

Quarlep
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Lets suppose we have two inequalities,
First inequality is x-y≤a-b≤x+y
Second inequality is t-g≤c-d≤t+g How can I multiply these inequalities

Thanks
 
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What do you have in mind by multiplying inequalities? In any case if all the terms are positive then term by term multiplication is OK. Otherwise be very careful.
 
Thanks
 
If you do not know whether these expressions can be negative, you can't validly multiply inequalities into new inequalities.

If you DO know that all the 8 individual numbers are, say, positive, you may first rearrange your inequalities, to for example:
x+b<=a+y<=x+2y+b and THEN multiply with the similary rearranged second inequality.
 

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