Why Do You Flip the Inequality Sign When Multiplying by a Negative?

Watch it happen with real numbers

Before any rule is stated, do the arithmetic and look at what comes out. Take a true statement, do the same thing to both sides, and check whether it is still true.

The numbers do the arguing. Nothing has been decided by convention here - in the second, third and fourth lines the statement produced is simply false unless the sign is turned round.

• 3 < 5, multiply both by 2 → 6 and 10, and 6 < 10. Order unchanged.
• 3 < 5, multiply both by -1 → -3 and -5, and -3 > -5. Order reversed.
• 3 < 5, multiply both by -2 → -6 and -10, and -6 > -10. Order reversed.
• 8 > 2, divide both by -2 → -4 and -1, and -4 < -1. Order reversed.

What multiplying by a negative does to the number line

Multiplying by -1 reflects every point through zero. A number three units to the right lands three units to the left, and one ten units to the right lands ten units to the left. Reflection keeps distances the same and reverses order - whatever was further right is now further left. That is the entire mechanism.

Multiplying by -2 does two things at once: it reflects and it stretches. The stretch changes the gaps between the numbers but not their order, so only the reflection is doing the reversing. This is why the size of the negative number is irrelevant to the rule. Multiplying by -0.1 flips the sign exactly as multiplying by -100 does.

It also explains the confusion underneath the whole topic, which is that -10 looks bigger than -6 and is smaller. The honest picture is a thermometer: -10 °C is colder than -6 °C, so it is the lower number, even though 10 is more than 6. Students who are secure about that rarely have trouble with the flip.

The moves that leave the sign alone

The rule is narrow, and applying it too widely is as costly as forgetting it. Everything in the list below preserves the direction of the sign.

The last line is worth separating out because it produces the same wrong answer as forgetting to flip and is a completely different error. 5 > x and x < 5 say the same thing; reading a solution off the page from left to right without turning it round gives x > 5, which is wrong. If you have solved an inequality and the unknown ended up on the right, rewrite the whole statement before you write your final answer.

• Adding the same number to both sides, positive or negative: x + 4 < 9 gives x < 5
• Subtracting the same number from both sides: x - 3 ≥ 2 gives x ≥ 5
• Multiplying or dividing both sides by a positive number: 4x < 20 gives x < 5
• Swapping the two sides - but then the sign must be turned round because the reader is now approaching it from the other end: 5 > x is the same statement as x < 5

The one move you must not make

Never multiply or divide both sides by an expression containing the unknown, because you do not know its sign and therefore do not know whether to flip. In 6/x > 2, multiplying both sides by x looks harmless and is not: x = 1 satisfies the original inequality and x = -1 does not, so any single answer produced by that move is wrong for half the number line.

The safe route is to avoid dividing by a negative in the first place. Where the unknown has a negative coefficient, move it to the other side so that it becomes positive. From -3x > 12, adding 3x to both sides and subtracting 12 gives 3x < -12, and dividing by the positive 3 gives x < -4 with no flip needed anywhere. Check it with a value: x = -5 gives -3 × -5 = 15, which is indeed greater than 12.

For quadratic inequalities - x² - 3x - 10 < 0 and its relatives, which appear on GCSE Higher and IGCSE Extended - neither route works. Factorise, find where the expression is zero, sketch the parabola, and read off the region where the curve is below or above the axis. The sketch replaces the sign rule entirely, which is why it is the method every mark scheme is built around.