What is a linear inequality?
An inequality says one side is bigger or smaller than the other. The signs are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to).
- Strict inequalities use < or >. Slack (non-strict) ones use ≤ or ≥.
- Linear in one variable means only one letter, with power 1: 3x − 5 < 7 or 2(x − 1) ≥ x + 4.
- The solution set is the set of all values that make it true. Usually it has infinitely many numbers.
Rules for solving inequalities
- Rule 1: You may add or subtract the same number on both sides. The sign does not change.
- Rule 2: You may multiply or divide both sides by the same positive number. The sign does not change.
- Rule 3: If you multiply or divide both sides by a negative number, you must reverse the sign.
Why Rule 3? 2 < 5. Multiply by −1: −2 and −5. On the number line, −2 is to the right of −5, so −2 > −5. Multiplying by a negative mirrors the line and reverses the order (step 3 of the 3D).
Never multiply both sides by a variable like x, because you do not know if x is positive or negative.
Solving in one variable: step by step
Solve 5 − 2x ≤ 11.
- Subtract 5: −2x ≤ 6.
- Divide by −2 (negative, so flip): x ≥ −3.
- Check with x = 0: 5 − 0 = 5 ≤ 11 ✓. Check with x = −4: 5 + 8 = 13 ≤ 11 ✗. So values below −3 fail, as expected.
The domain matters. If x is a natural number, x < 4 means x = 1, 2, 3. If x is an integer, it means …, −1, 0, 1, 2, 3. If x is real, it means every number less than 4.
Showing the solution on a number line
- Draw the number line and mark the end number.
- Use a hollow (open) dot for < or >: the end is not included.
- Use a filled (closed) dot for ≤ or ≥: the end is included.
- Shade to the left for < or ≤ and to the right for > or ≥.
In interval notation: x < 4 is (−∞, 4); x ≥ −3 is [−3, ∞); −1 ≤ x < 3 is [−1, 3). A round bracket means ‘not included’ and a square bracket means ‘included’.
For integer or natural answers, draw only separate dots, not a shaded bar (step 5 of the 3D).
Double inequalities and systems
A double inequality like −5 ≤ 2x + 1 < 7 is solved by doing the same thing to all three parts: subtract 1 → −6 ≤ 2x < 6; divide by 2 → −3 ≤ x < 3.
For a system of two inequalities, solve each one, draw both on the same line, and take the overlap. Example: x > −2 and x ≤ 5 give −2 < x ≤ 5. If there is no overlap, the system has no solution.
Word problems and board exam focus
Key phrases: ‘at least’ means ≥, ‘at most’ or ‘not more than’ means ≤, ‘more than’ means >, ‘less than’ means <.
Common board questions: solve and graph (2–3 marks), find the natural/integer solutions (1–2 marks), a word problem about marks, cost or temperature (3–4 marks). Always state the solution set and draw the line.
Key formulas and definitions
- a < b ⇒ a + c < b + c and a − c < b − c
- a < b, k > 0 ⇒ ka < kb and a/k < b/k
- a < b, k < 0 ⇒ ka > kb and a/k > b/k (sign flips)
- Hollow dot: <, >. Filled dot: ≤, ≥
- x < a ↔ (−∞, a); x ≥ a ↔ [a, ∞); a ≤ x < b ↔ [a, b)
Worked examples
1. Solve 4x < 20 when x is (i) a natural number (ii) a real number.
Divide by 4 (positive): x < 5. (i) Natural numbers: {1, 2, 3, 4}. (ii) Real numbers: every x less than 5, the interval (−∞, 5), shown with a hollow dot at 5 and shading to the left.
2. Solve 3x − 7 > 2 and show it on a number line.
Add 7: 3x > 9. Divide by 3: x > 3. Hollow dot at 3, shade to the right. Solution (3, ∞).
3. Solve 5 − 2x ≤ 11.
Subtract 5: −2x ≤ 6. Divide by −2 and flip the sign: x ≥ −3. Filled dot at −3, shade right. Solution [−3, ∞).
4. Solve (x − 2)/3 ≥ (x + 1)/4.
Multiply both sides by 12 (positive): 4(x − 2) ≥ 3(x + 1). So 4x − 8 ≥ 3x + 3. Subtract 3x and add 8: x ≥ 11. Solution [11, ∞).
5. Solve −4 < 3 − 2x ≤ 9.
Subtract 3 from all parts: −7 < −2x ≤ 6. Divide all parts by −2 and flip both signs: 7/2 > x ≥ −3. So −3 ≤ x < 3.5, the interval [−3, 3.5).
6. Solve the system 2x + 1 > 5 and 3x − 4 ≤ 11.
First: 2x > 4, x > 2. Second: 3x ≤ 15, x ≤ 5. Overlap: 2 < x ≤ 5, hollow dot at 2 and filled dot at 5 with the part between shaded.
7. Ria scored 62 and 70 in two tests. What must she score in the third test (out of 100) to have an average of at least 65?
Let x be the third score. (62 + 70 + x)/3 ≥ 65, so 132 + x ≥ 195, which gives x ≥ 63. Since marks cannot exceed 100, 63 ≤ x ≤ 100.
8. Find all pairs of consecutive odd natural numbers, both greater than 10, whose sum is less than 40.
Let them be x and x + 2 with x odd and x > 10. x + x + 2 < 40 gives x < 19. Odd x with 10 < x < 19: 11, 13, 15, 17. Pairs: (11, 13), (13, 15), (15, 17), (17, 19).
Common mistakes
- Forgetting to flip the sign when dividing by a negative number, like −3x < 9 → x < −3 (wrong). Correct: x > −3.
- Using a filled dot for < or >. Filled dots are only for ≤ and ≥.
- Multiplying both sides by x or by an expression whose sign you do not know.
- Shading a whole bar when x must be a natural number or integer. Draw separate dots instead.