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Multiplication and division inequalities, step by step

Multiplying or dividing an inequality by a negative flips the sign.

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The question

−3x < −9

−3x < −9
-9÷−3=3
<→>
x>3
  1. An inequality comes apart exactly like an equation — the same moves, done to both sides. There is one extra rule and it arrives at the end, so start by ignoring the sign and solving for x.
  2. Now divide both sides by −3: -9 ÷ −3 = 3.
  3. Here is the extra rule, and it is the only one: dividing both sides by a NEGATIVE turns the sign round, so < becomes >. Do not take that on trust — put a number from the answer back into the question and see which way it has to face.
  4. So x > 3. That is a whole RANGE of numbers rather than one, which is what makes this different from an equation — and why the sign is half the answer.

How it works.

  1. 01

    An inequality comes apart exactly like an equation — the same moves, done to both sides. There is one extra rule and it arrives at the end, so start by ignoring the sign and solving for x.

  2. 02

    Now divide both sides by −3: -9 ÷ −3 = 3.

  3. 03

    Here is the extra rule, and it is the only one: dividing both sides by a NEGATIVE turns the sign round, so < becomes >. Do not take that on trust — put a number from the answer back into the question and see which way it has to face.

  4. 04

    So x > 3. That is a whole RANGE of numbers rather than one, which is what makes this different from an equation — and why the sign is half the answer.

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