Theorem

Multiplying an inequality by a negative number reverses it

Algebra • Expressions • Inequalities

Formula

a < b and c < 0 \Rightarrow a c > b c

For real

a

b

c

: if

a < b

and

c > 0

then

a c < b c

, but if

a < b

and

c < 0

then

a c > b c

. So multiplying or dividing both sides of an inequality by a negative number reverses the sign, while adding or subtracting any quantity leaves it alone; multiplying by

0

destroys the inequality altogether.

How to use it

1Settle the sign of the coefficient before dividing an inequality through by it, and flip the sign if it is negative.
2Taking reciprocals of two quantities of the same sign reverses the order too: $0 < a < b$ gives $\frac{1}{a} > \frac{1}{b}$ .
3If the multiplier contains the unknown, as $x$ does in $\frac{x - 2}{x} < 1$ , do not multiply through at all — split into the cases $x > 0$ and $x < 0$ , or rearrange and compare with $0$ .

Worked example

Solve

3 - 2 x \leq 9

for real

x

Step-by-step

1
$3 - 2 x \leq 9 \Rightarrow - 2 x \leq 6$
Why: Subtracting $3$ from both sides does not affect the direction.
2
Divide by $- 2$ : $x \geq - 3$
Why: The divisor is negative, so the sign reverses — this is the step the rule governs.

Answer

x \geq - 3

Watch out

The reversal applies ONLY to multiplication and division by a negative, never to addition or subtraction. Multiplying through by an expression whose sign is unknown, such as

x

x - 1

, is not permitted in either direction, and treating it as positive by default is the commonest way an inequality question is lost.

Also known as

reverse the inequality sign, multiply inequality by a negative, dividing an inequality by a negative number, flip the inequality, inequality sign change

Multiplying an inequality by a negative number reverses it

Formula

Worked example

Appears in past papers

2020 (II)

2020 (I)