Solving and representing linear inequalities
Solving an inequality is almost identical to solving an equation — except for one critical rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign.
The four symbols
<"less than" — strict, does not include equality.>"greater than" — strict.≤"less than or equal to" — includes the boundary value.≥"greater than or equal to" — includes the boundary.
The flip rule
-2x < 6. Divide both sides by -2 (negative): flip the sign. Result: x > -3.
This is the only rule that distinguishes inequalities from equations. Adding/subtracting any number, multiplying/dividing by a positive number — no flip needed.
Number-line representation
- Open circle ○ for strict inequality (
<,>). - Closed circle ● for non-strict inequality (
≤,≥). - Arrow points in the direction of valid x-values.
x ≥ 2: closed circle at 2, arrow pointing right.
x < -1: open circle at -1, arrow pointing left.
Compound inequalities
-3 < x ≤ 5 means x is greater than -3 AND less than or equal to 5. On a number line: open circle at -3, closed at 5, segment between them shaded.
✦Worked example— Worked example 1 — basic
Solve 3x + 4 ≤ 19.
Subtract 4: 3x ≤ 15. Divide by 3: x ≤ 5. (No flip — divided by positive.)
✦Worked example— Worked example 2 — flip
Solve -2x + 7 > 1.
Subtract 7: -2x > -6. Divide by -2 AND flip: x < 3.
Always check: try x = 0 (which is in the proposed solution): 0 + 7 = 7 > 1 ✓.
✦Worked example— Worked example 3 — variable on both sides
5 - 2x ≤ 11 - 4x. Add 4x: 5 + 2x ≤ 11. Subtract 5: 2x ≤ 6. Divide: x ≤ 3.
✦Worked example— Worked example 4 — listing integer solutions
"Find the integer values of n satisfying -3 < 2n + 1 ≤ 9."
Subtract 1 from each part: -4 < 2n ≤ 8. Divide by 2: -2 < n ≤ 4.
Integers: n = -1, 0, 1, 2, 3, 4 — six values.
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting to flip when multiplying/dividing by a negative.
- Using the wrong type of circle. Open for strict; closed for non-strict.
- Listing endpoint that doesn't qualify.
-2 < n ≤ 4— n = -2 is excluded (strict). n = 4 is included. - Misreading "between". "Between 3 and 7" usually means
3 < x < 7(strict), but exam wording matters — check. - Inequalities of two ends with single variable.
-3 < x ≤ 5is shorthand for two inequalities — solve them as one block.
➜Try this— Quick check
Solve 4 − 3x ≥ 13. Subtract 4: −3x ≥ 9. Divide by −3 and flip: x ≤ −3.
AI-generated · claude-opus-4-7 · v3-deep-algebra