OK, teachers, homeschooling parents and tutors … raise your hand if you’ve ever felt uncomfortable when students pose that question about inequalities?
That question being: why do we flip the inequality symbol when we multiply or divide by a negative number?
I’d have to, sheepishly, raise my hand.
So when I got asked that question once again last week, I decided to figure it out and come up with an answer that would help students understand this point.
What I came up with is that it’s easiest to explain this through a combination of examples and logic. First, the examples.
Let’s break the situation up into three cases. We could have inequalities in which the numbers on the two sides are A) both positive, B) both negative, or C) one number positive, the other negative.
Let’s start with Case A. Suppose we start with the statement, 2 < 4
Now, multiply both sides by a positive number (let’s use 3), and we get: 6 < 12. Still true, right?
But take the original inequality and multiply it by a negative number (let’s use – 3), and we get: – 6 < – 12. Not true, right? But if we flip the sign, we do get a true statement: – 6 > – 12
Case B. Now let’s start with two negative numbers in our true inequality: – 4 < – 2 If we multiply both sides by a positive number (3 again), we get: – 12 < – 6, which is again true.
But if we multiply this inequality by a negative number (– 3 again), we get: 12 < 6, which is obviously false. However if we once again flip the sign, we get a true statement: 12 > 6.
Finally, Case C. Now we start with an inequality that has both a positive and a negative number: – 2 < 4. If we multiply both sides by positive 3, we get: – 6 < 12, which is still true.
But if we multiply both sides by our – 3 again, we get: 6 < – 12, which is once again false. And again we need to flip the sign to make it true: 6 > – 12.
So far so good, but this lacks the logic of an explanation. How can we bring in some logic and reasoning, to help students see why all of this stuff happens?
Here’s my — granted, informal — way of explaining this. When we multiply or divide a number by positive numbers, we don’t change its sign; if the number was positive, it stays positive, and if it was negative, it stays negative. But when we multiply or divide a number by a negative number, we do change its sign … either from positive to negative, or from negative to positive.
So the reason that we flip the inequality symbol must be related to the fact that — by multiplying or dividing both sides of the inequality by a negative number — we are changing the signs of both numbers in the inequality. But how exactly does this work?
The answer, it turns out, is rooted in the relationship between the absolute value of numbers and their relative sizes. For numbers that are positive, there’s one way to tell which number is larger … the number with the larger absolute value is the larger number. For example, comparing 4 and 12, we know that 12 is larger than 4 because the absolute value of 12 is larger than the absolute value of 4. But for numbers that are negative, the exact opposite is true. For two negative numbers, the number with the larger absolute value is actually the smaller number. For example, compare – 4 and – 12. Their absolute values are 4 and 12, respectively, but the number with the larger absolute value is in fact the smaller number, not the larger number. In this example, – 12 (with the bigger absolute value of 12), is in fact smaller than – 4 (with the smaller absolute value of 4).
So the point to remember here is that there are two different relationships between the absolute values of numbers and the relative sizes of numbers. For positive numbers, the greater the absolute value, the greater the number. But for negative numbers, the greater the absolute value, the smaller the number.
This fact has an impact on inequalities where we change the signs of the numbers. Before changing the signs of the numbers, the numbers on the two sides of the inequality had one size relationship; one number was larger than the other (let’s say that Number A is, at this stage, larger than Number B). But when we multiply or divide both of these numbers by a negative, we flip the signs of both numbers. And by flipping the signs of both numbers, we change the size relationship of the numbers to each other. The one that was the larger one ends up being the smaller one, and vice-versa. So in our abstract example, if Number A was larger than Number B before their signs were changed, after both signs are changed, Number A will be smaller than Number B.
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!