Why Flip the Inequality Sign?
Date: 10/26/2001 at 11:30:05 From: Sean Subject: Flipping the Inequality sign I have a question about an inequality problem: Solve and graph 5 - 3x => 17. (=> is greater than or equal to). Please tell me why you flip the inequality sign when dividing by a negative number. Thanks. Sean
Date: 10/26/2001 at 12:09:56 From: Doctor Peterson Subject: Re: Flipping the Inequality sign Hi, Sean. Let's try solving your inequality in two ways, one dividing by a negative number and the other avoiding that. First, 5 - 3x >= 17 -3x >= 12 x <= -4 Here I subtracted 5, then divided by -3 and reversed the inequality. Next, 5 - 3x >= 17 5 >= 17 + 3x -12 >= 3x -4 >= x This time I added 3x to both sides, eliminating the negative coefficient; then I subtracted 17 and divided by 3. Notice that this gives the same result, though it's written backwards. We know there's nothing tricky in this method; so the reversal of the inequality must be right when we do it the other way. We can prove the general rule the same way: if a >= b then a - b >= 0 (subtracting b from both sides) and -b >= -a (subtracting a from both sides) so that -a <= -b (rewriting the inequality in reverse) This shows that when you multiply an inequality by -1 (which is part of what you do when you divide by a negative number), you reverse the inequality. Here are some other discussions of this from our archives, which I found by searching for the words "reverse inequality negative": Negatives and Inequalities http://mathforum.org/library/drmath/view/53142.html Reversing the Inequality http://mathforum.org/library/drmath/view/57463.html Inequalities and Negative Numbers http://mathforum.org/library/drmath/view/53177.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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