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Multiplication and division inequalities worksheets

Multiplying or dividing an inequality by a negative flips the sign — the one rule that separates inequalities from equations, and where most marks are lost. These free printable worksheets drill exactly that.

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Multiplication and division inequalities
Multiplication and division inequalities worksheet — One step, sign included, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

3x < 12 is solved like an equation: divide both sides by 3 to get x < 4, sign unchanged. But −2x < 8 divides by a negative, so the sign turns over — x > −4, not x < −4. Both kinds sit on every sheet, because a page of only positives never triggers the flip and teaches a rule that is silently wrong the first time a negative coefficient appears.

Why this sheet works

The answer includes the sign, so a solution with the right number and the wrong direction is marked as wrong — which is the only way the flip is actually learned. It is worth testing rather than memorizing: put a number from the range back into the original and see whether it holds. This is the multiplication-and-division cut of solving inequalities, kept apart from the addition-and-subtraction one so the flip gets its own page.

How this one works

One question from this sheet, worked through a step at a time.

−4x > 44

−4x > 44
44÷−4=−11
>→<
x<−11
  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 −4: 44 ÷ −4 = −11.
  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 < −11. 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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