The Flip Rule — Why a Negative Reverses the Sign

In this lesson you will: Reverse the inequality sign when you multiply or divide both sides by a negative number; Explain the reversal using the number line rather than quoting it as a rule; Recognize the moves that never require a reversal.

Watch what a negative does to order

Start with something nobody disputes: -2<3. Now multiply both numbers by -2. The -2 becomes 4. The 3 becomes -6.

Look at where they landed. The number that was on the left is now on the right. 4>-6. The relationship did not disappear — it turned around.

Multiplying by a negative reflects the line

Multiplying every number by a negative flips the whole number line across zero. Whatever was far left lands far right, and whatever was far right lands far left. A reflection reverses left-to-right order, and left-to-right order is exactly what an inequality sign reports.

Dividing by a negative is the same thing, since dividing by -4 is multiplying by -1/4. So both operations reverse the sign, and for the same reason.

What does not flip

Adding or subtracting never reverses anything, no matter how many minus signs are in sight. Sliding both numbers three units left keeps the left one on the left.

Multiplying or dividing by a positive number does not reverse anything either. Stretching the line away from zero moves numbers apart but does not reorder them.

Common mistake — Flipping because a minus sign appeared somewhere

A student solves x-8>-2, correctly reaches x>6, then reverses it to x<6 \"because of the negative.\"

The instinct is trained, not stupid: the flip rule gets taught as \"negatives reverse the sign,\" and there is a negative right there in the problem. The rule as remembered says nothing about which negative.

But the step performed was adding 8 to both sides. Addition never reverses an inequality. The -2 was a term, not a multiplier.

The mirror-image error is worse and more common: solving -4x>12 and writing x>-3, forgetting the reversal on the step that genuinely needs it.

How to catch it: Name the operation you performed on both sides, out loud if necessary. Did you multiply or divide both sides by a negative number? If not, nothing reverses. Then test a value: x=7 in x-8>-2 gives -1>-2, true, so x>6 is right and x<6 is not.