Reverse the Direction when Dividing by a Negative

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve −2x < -4. Candidate answers: (A) x < 2 (B) x = 2 (C) x > -2 (D) x > 2.
• Independent check inputs: Solve −2x < -6. Candidate answers: (A) x < 3 (B) x = 3 (C) x > -3 (D) x > 3.

Learn the idea

Solving means finding inputs that make a relationship true. Balanced operations preserve equality, whereas inequality direction must reverse when multiplication or division by a negative number reverses number-line order. Multiplying an ordered pair of numbers by a negative reflects their positions across zero. The larger value becomes the smaller, which explains the inequality reversal rule.

Here is the method on the supplied teaching case. Dividing both sides by −2 reverses the inequality direction. The boundary is -4/(−2)=2; equality is excluded.

Reverse the sign only when multiplying or dividing both sides by a negative. Adding a negative does not cause reversal; a strict inequality also excludes its boundary point.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Solve −2x < -18. Candidate answers: (A) x < 9 (B) x = 9 (C) x > -9 (D) x > 9. Show the mathematical reason, not only the selected answer. Expected result: x > 9. Dividing both sides by −2 reverses the inequality direction. The boundary is -18/(−2)=9; equality is excluded.

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