Inequalities • Topic 2 of 4

Flipping the Sign

Here is the one rule that separates inequalities from equations: whenever you multiply or divide both sides by a negative number, the inequality symbol reverses. −2x < 6 becomes x > −3, not x < −3, because dividing by −2 flips the <. The logic is easy to see with numbers: 3 < 5 is true, but multiply both sides by −1 and you get −3 and −5, where −3 > −5. Adding or subtracting a negative never triggers the flip — only multiplying or dividing by one does. A safe habit that avoids the trap entirely is to move the variable term to whichever side keeps its coefficient positive, so you never divide by a negative in the first place. This flip is a favourite GRE trap, especially embedded in Quantitative Comparison where the direction of the inequality is the entire answer.

✅ Solved examples

1. Solve −3x > 12.
Divide by −3 and flip: x < −4.
2. Solve 8 − 2x ≥ 14.
Subtract 8: −2x ≥ 6. Divide by −2 and flip: x ≤ −3.
3. Solve 5 − x < 9 two ways.
Subtract 5: −x < 4, divide by −1 and flip: x > −4. (Or add x: 5 < 9 + x ⇒ x > −4 — no flip needed.)
4. Solve 4 − 3x ≤ 2x − 6.
Collect to keep x positive: 4 + 6 ≤ 2x + 3x ⇒ 10 ≤ 5x ⇒ x ≥ 2 (no flip, because we moved −3x across).

✏️ Practice — try these, take hints as needed

1. Solve −4x ≤ 20.
Divide by −4.
Flip the sign.
x ≥ −5.
x ≥ −5
2. Solve 10 − 5x > 0.
Subtract 10: −5x > −10.
Divide by −5 and flip.
x < 2.
x < 2
3. Solve 3 − 2x ≥ 11.
Subtract 3.
−2x ≥ 8.
Divide by −2, flip.
x ≤ −4
4. Solve 6 − x < 2x without dividing by a negative.
Add x to both sides.
6 < 3x.
Divide by 3.
x > 2
5. Solve −(x + 2) > 3.
Distribute: −x − 2 > 3.
−x > 5.
Multiply by −1 and flip.
x < −5

📝 Topic test — 8 questions

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