You have probably remembered in your high school or college days the following rule: If we multiply the numbers on both sides of the inequality by a negative number, then the inequality sign should be reversed.
For example, if we want to find the solution of the inequality , we multiply both sides by and reverse the greater than sign giving us . Now, why did the sign became ?
Before we proceed with our discussion, let us first remember 2 important things 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 and the positive numbers are at its right hand side such as shown in Figure 1.
- If we have numbers and , then if is at the right of on the number line. For example, in Figure 1, since is at the right of .
If we generalize the statements above, suppose we have two numbers, say, and such that , if we multiply them to a negative number , instead of having , the answer should be .
For specific values, let’s choose and as shown in the diagram above and choose . Note that we will just use these values for discussion purposes, but we may take any value, as long as the conditions above hold. It would help, if we think of and as two points on the number line with as a blue point on the right , a red point.
And note that before multiplying with a negative number, VALUE OF BLUE POINT > VALUE OF RED POINT.
Since and are variables, we need to multiply all the numbers on the number line by . This is to ensure that whatever values we choose for and , we multiplied it by . If we multiply every number on the number line by , the geometric consequence would be a number line with negative numbers on the right hand side of , and positive numbers at the left hand side of as shown in Figure 2.
But negative numbers should be at the left hand side of so we reverse the position of the number line, or rotate it 180 degrees with zero as the point of rotation. The resulting figure is shown in Figure 3.
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 and are real numbers, and , then .
Our summary above is actually a mathematical theorem. The proof of this theorem is shown below. It is a very easy proof, so, I suppose, that you would be able to understand it.
Theorem: If and are real numbers, with and , then .
Subtracting from both sides, we have .
Now, means is positive.
Since is negative, therefore, is negative (negative multiplied by positive is negative)
Since is negative, therefore, .
Distributing , we have .
Adding to both sides, we have which is what we want to show .