Multiplying by x when you do not know its sign.

Graphs of y = 1/x and y = 2 (or a chosen constant k) with the solution of 1/x < k shaded on the x-axis; a 'multiply by x' button shows the naive answer and highlights the negative x it loses.

For 1/x < 2, shade the full solution set (x < 0 or x > 1/2) and spot the region the naive answer misses.

Multiplying 1/x < 2 by x without knowing the sign of x.

• Rule 1: adding the same number to both sides, or subtracting the same number from both sides, keeps the inequality true with the same sign.
• Rule 2: multiplying or dividing both sides by the same positive number keeps the sign. Multiplying or dividing both sides by a negative number reverses the sign: < becomes >, ≤ becomes ≥, and so on.
• Why the reversal: 5 > 3, but −5 < −3. Multiplying by −1 reflects numbers through 0 on the number line, so their order swaps. Similarly −6 < −4, yet (−6)(−2) = 12 > 8 = (−4)(−2).
• Worked example: 7x − 4 < 4x + 8. Subtract 4x: 3x − 4 < 8. Add 4: 3x < 12. Divide by 3 (positive): x < 4. Check: x = 3 gives 17 < 20 (true) and x = 5 gives 31 < 28 (false).
• Worked example with a reversal: 2x + 5 > 5x − 10. Subtract 5x: −3x + 5 > −10. Subtract 5: −3x > −15. Divide by −3 and reverse: x < 5. Check: x = 4 gives 13 > 10 (true).
• Worked example: −3x ≥ 12. Divide by −3 and reverse: x ≤ −4. Check: x = −5 gives 15 ≥ 12 (true) and x = 0 gives 0 ≥ 12 (false).
• One safe habit: gather the x-terms on the side where their coefficient stays positive. For 2x + 5 > 5x − 10, moving 2x to the right gives 15 > 3x, so 5 > x with no reversal needed.
• Never multiply or divide by an expression whose sign is unknown, such as x itself, without splitting into cases; the direction of the sign would be unknown.

Teaching this idea? Put the simulation on your own page or LMS with this code.