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Why do we reverse/flip the inequality sign?

October 15, 2009 GB High School Algebra, High School Mathematics, Questions and Quandaries

You have probably remembered in Algebra that if we multiply an inequality by a negative number, then the inequality sign should be flipped or reversed. For example, if we want to find the solution of the inequality $-\frac{1}{2}x > 8$ , we multiply both sides by $-2$ and reverse the greater than sign giving us $x < -16$ . Now, why did the $>$ sign became $<$ ?

If we generalize the statements above, suppose we have two numbers, say, $a$ and $b$ such that $a > b$ , if we multiply them to a negative number $c$ , instead of having $ac > bc$ , the answer should be $ac < bc$ .

Before we proceed with our discussion, let us first remember 2 basic concepts we have learned in elementary mathematics:

The number line is arranged in such a way that the negative numbers are at the left hand side of $0$ and the positive numbers are at its right hand side such as shown in Figure 1.
If we have $2$ numbers $a$ and $b$ , then $a > b$ if $a$ is at the right of $b$ on the number line. For example, in Figure 1, $2 > -1$ since $2$ is at the right of $-1$ .

Figure 1 – The number line

For specific values, let’s choose $a = 2$ and $b = -1$ as shown in the diagram above and choose $c = -1$ . Note that we will just use these values for discussion purposes, but we may take any values. It would help, if we think of $a$ and $b$ as two points on the number line with $a$ as a blue point on the right $b$ , a red point.

And note that before multiplying with a negative number, VALUE OF BLUE POINT > VALUE OF RED POINT.

Since $a$ and $b$ are variables, we need to multiply all the numbers on the number line by $-1$ . This is to ensure that whatever values we choose for $a$ and $b$ , we multiply them by $-1$ . If we multiply every number on the number line by $-1$ , the geometric consequence would be a number line with negative numbers on the right hand side of $0$ , and positive numbers at the left hand side of $0$ as shown in Figure 2.

Figure 2 – Afer multiplying all numbers on the number line by -1

But negative numbers should be at the left hand side of $0$ so we reverse its position by rotating it 180 degrees from any point of rotation (for example, 0). The resulting figure is shown in Figure 3.

Figure 2 - All numbers in the number line were multiplied by -1

Notice that the blue and red points changed order and that the blue point is now at the left of the red point. Therefore, VALUE OF BLUE POINT < VALUE OF RED POINT. That is, why the inequality sign was reversed.

Summarizing, multiplying an inequality by a negative number is the same as reversing their order on the number line. That is, if $a, b$ and $c$ are real numbers, $a > b$ and $c<0$ , then $ac < bc$ .

Our summary above is actually a mathematical theorem. The proof of this is shown below. It is a very easy proof, so, I suppose, that you would be able to understand it.

Theorem: If $a, b$ and $c$ are real numbers, with $a > b$ and $c<0$ , then $ac < bc$ .

Proof:

Subtracting $b$ from both sides, we have $a - b>0$ .