Negatives and InequalitiesDate: 09/10/98 at 19:27:52 From: Prerana Jain Subject: Inequalities - why switch the sign? Could you please explain to me why when we multiply or divide an inequality by a negative number, we have to reverse the inequality sign? Thank you. Date: 09/10/98 at 20:08:58 From: Doctor Sam Subject: Re: Inequalities - why switch the sign? Prerana, Good question. It is puzzling why the sign should switch. Here is one way to think about it. Multiplying a quantity by -1 changes it into its opposite. For example, 3 becomes the opposite of 3 or -3, and 12 becomes the opposite of 12, which is -12. Now think about how these numbers fall on a number line: ----------------------0----1----2----3--------------------12----- Since 3 < 12, three is closer to zero and twelve is farther away. That's true in general. Numbers with larger magnitudes are farther away from zero. Now what happens if we take opposites? That is, if we multiply or divide these numbers by -1? Opposites are the same distance from zero so -12 will be farther away from 0 than -3: --- -12 --------------- -3 ----- 0 --------------------------- This is true in general. If A < B and both are positive, then the opposite of B will be farther to the left than the opposite of A, that is, -B < -A which means the same thing as -A > -B. If you just look at the symbols it seems as if we put minus signs in front of each letter and switch the inequality symbol. But if you look at the meaning of the symbols on the number line it is much clearer: -B -A 0 A B --------------------------------------------------------- If you look at a number line picture for A < B where both numbers are negative instead of positive (for example -4 < -1) you will see the same pattern occurs when you take the opposite of each. The final case is A < B where A is negative and B is positive. You can draw a number line picture for this case too, but it is almost simpler to just think about the meaning of the opposites. If A is negative, then its opposite, -A, is a positive number. If B is positive, then its opposite, -B, is a negative number. Of course, any negative number is smaller than any positive number so -B < -A. I hope that helps. - Doctor Sam, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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