Euler Learn

Why does multiplying an inequality by a negative reverse the sign?

A proof using the difference between the two sides, a number-line interpretation, and a checkable example.

If $a<b$ and $c<0$ , then $ca>cb$ . Multiplication by a negative combines a change of scale with reflection across zero, and that reflection reverses the order.

A proof from the difference between the sides

From $a<b$ , the difference $b-a$ is positive. If $c<0$ , then $-c>0$ , so the product of two positive numbers is positive: $(-c)(b-a)>0.$ Expanding gives $ca-cb>0.$ This says exactly that $ca>cb$ . The direction changes because that is the order relation that remains true after the multiplication.

What happens on the number line

Multiplication by $-1$ reflects every point across zero. For example, $-2<5$ , but after reflection we have $2>-5$ . Points that were farther right move farther left, and vice versa. Multiplication by any negative number, such as $-3$ , adds a positive scale factor to the same reflection: from $-2<5$ we obtain $6>-15$ .

A checkable example: solve −4x + 3 ≤ 19

Start with $-4x+3\le19.$ Subtracting $3$ from both sides gives $-4x\le16.$ Now divide by $-4$ . Because the divisor is negative, reverse the direction to obtain $x\ge-4.$ Check the boundary: at $x=-4$ , both sides equal $19$ . An included value, $x=0$ , gives $3\le19$ ; an excluded value, $x=-5$ , gives $23\le19$ , which is false. These checks confirm $x\ge-4$ .

Division, zero, and multipliers with an unknown sign

Division by a negative reverses the direction for the same reason: it is multiplication by a reciprocal that is also negative. The guide to dividing by a fraction derives that inverse operation.

Multiplication by zero does not give an equivalent inequality: both sides become zero, so the order information disappears. If the multiplier’s sign is unknown, split the work into positive, negative, and zero cases. To test a specific exercise, ask the Euler tutor to check each step.

Frequently asked questions

Does adding or subtracting a negative number reverse the direction?

No. Adding the same number to both sides translates both points by the same amount and preserves their order.

Does the rule also apply to ≤ and ≥?

Yes. When both sides are multiplied or divided by a negative, ≤ becomes ≥ and ≥ becomes ≤; equality at the boundary remains included.

What if I need to multiply by a variable?

If the variable’s sign is unknown, separate the cases: a positive value preserves the direction, a negative value reverses it, and zero may erase the order information.

How can I catch a forgotten sign reversal?

Test one value that should lie inside the solution set and one that should lie outside. Substitution into the original inequality quickly exposes the wrong direction.