Why do we reverse/flip the inequality sign?

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 -\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 <?

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

  1. 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.
  2. 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.
Number line

Figure 1 – The number line

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.

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 value, as long as the conditions above hold. 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 multiplied it 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.

num4-1

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 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.

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 POINTVALUE 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 theorem 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.

Now, a - b>0 means a - b is positive.

Since c is negative, therefore, c(a - b) is negative (negative multiplied by positive is negative)

Since c(a - b) is negative, therefore, c(a - b) < 0.

Distributing c, we have ac - bc < 0.

Adding bc to both sides, we have ac < bc which is what we want to show .\blacksquare

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4 Comments

  1. Thank you for explaining this, I had problems with this but I get it now. I have one question, what happens if both a and b are negative? Would it still reverse the inequality sign? Thank you~

    Reply
    • @appplebananafruits123

      If you are referring to the proof, you would reverse it if a and b are negative, but multiplying them by a negative value that is c will make them positive, so no need to reverse them. If you are however referring to inequalities like -3x < -5, then yes. The answer to that is x > 5/3.

      Reply

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