Why the Sign Flips
Drag the boundary k and the multiplier m. The top ray is every x greater than k; the bottom one is that same set, every point multiplied by m. Nothing here rewrites a symbol — watch what the ray itself does when m turns negative.
x > 3, multiplied by −2, becomes x < −6
m is negative, so multiplying reverses order.
- k
- 3
- m
- −2
- Sign flips?
- Yes
- Result
- x < −6
Try a multiplier
Ready to try some? Multiplication and division inequalities worksheets
Where it usually goes wrong
"Flip the sign when you multiply or divide by a negative" is memorized as a step to insert into a calculation, disconnected from what an inequality actually claims — that one set of numbers is bigger than another. So the rule survives being applied correctly for years without the student ever seeing the moment it describes: the set itself turning around.
Try this
- Press "× 2". The bottom ray still opens the same direction as the top one — it moved and stretched, but multiplying by a positive keeps every order relationship exactly as it was.
- Press "× −2". The bottom ray now opens the OPPOSITE way. Nothing about the symbol was touched by hand here — the ray is drawn from the actual transformed set, and it simply came out facing the other direction.
- Drag k back and forth with m still negative. The bottom ray's direction never changes back — only a negative m turns it, and it stays turned for every boundary.
Where this goes next
This is the same fact that makes −x < −y equivalent to x > y: multiplying both sides by −1 is exactly this transformation, and the inequality symbol is required to flip for the identical reason the ray in this picture does — order reverses under a negative multiplier, always, not just when a rule says to remember it.
More about numbers and proportion
- Cut Once, Counted Twicea(x + c) = ax + ac because cutting a rectangle's width and adding the two pieces' areas back together is not a special property of multiplication — it is what area already means
- Two Points, One Distancean absolute value equation has two solutions because they are the two points at the same distance from the center, not a positive answer and a negative one
- Up, Then Down, Not Evenincreasing a value by a percent and then decreasing the result by the same percent never returns to the start, because the decrease is taken from a bigger number than the increase was