Inequalities: when to flip the sign

Multiply or divide by a negative, and the inequality turns around.

The question

Solve −2x > 6

The trick

Solve it exactly like an equation, with one extra rule: dividing or multiplying by a negative number reverses the direction of the sign.

Worked example

1Divide both sides by −2
2Because −2 is negative, the > becomes <
3x < −3

So: x < −3

Remember it like this

A negative turns the number line around, so the arrow has to turn around too.

In the exam

Adding or subtracting a negative does *not* flip the sign — only multiplying or dividing by one does. Test your answer with a single value: if x < −3, try x = −4 in the original and check it holds.

Why it works

3 is less than 5. Multiply both by −1 and you get −3 and −5 — but now −3 is the *larger* one. Negatives reverse the order of the number line, so the symbol must reverse with them.

Explain this to my childGet the words to teach it, in your language

Try it yourself

0/3 correct

1
Solve −3x > 12. Is x < −4 or x > −4?
2
Solve 2x > 8. Is x > 4 or x < 4?
3
Solve −x < 5. Is x > −5 or x < −5?