Linear Inequalities & Interval Notation (Grade 10)

Solving linear inequalities, the one rule that catches everybody (dividing by a negative), showing the answer on a number line, and writing it in interval notation.

An inequality is solved almost exactly like an equation — with one extra rule. In Grade 10 you must also present the answer three ways: in symbols, on a number line, and in interval notation.

1The rule that catches everybody

Multiplying or dividing both sides by a negative number reverses the inequality sign.

Worked ExampleSolve: −3x > 12

1
Divide both sides by −3.
To get x on its own.
2
The sign flips: x < −4.
Dividing by a negative reverses > into <.
3
Check with x = −5: −3(−5) = 15 > 12 ✓
Always test one value from your answer.

⚠️

Forgetting to flip the sign is the single most common error in this topic. If your final step divides or multiplies by a negative, reverse the sign.

2Showing the answer on a number line

An open circle means the value is not included (< or >). A solid dot means it is included (≤ or ≥).

x ≥ 2 — solid dot because 2 is included, shaded to the right.

x < 1 — open circle because 1 is excluded, shaded to the left.

3Interval notation

[ ] square bracket → the endpoint is included (≤, ≥)
( ) round bracket → the endpoint is not included (<, >)
∞ always takes a round bracket.

x ≥ 2 → [2 ; ∞)
x < 1 → (−∞ ; 1)
−1 ≤ x < 3 → [−1 ; 3)

⚠️

Infinity is never "reached", so it can never take a square bracket. Writing [2 ; ∞] is always wrong.

4Compound inequalities

−1 ≤ x < 3 — solid dot at −1 (included), open circle at 3 (excluded), shaded between.

Worked ExampleSolve: −4 < 2x − 2 ≤ 6

1
Add 2 to all three parts: −2 < 2x ≤ 8.
Whatever you do, do it to every part.
2
Divide all three by 2: −1 < x ≤ 4.
2 is positive, so no sign flips.
3
Interval notation: (−1 ; 4].
Round at −1 (excluded), square at 4 (included).

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Solve: 2x + 3 < 11
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2x < 8 → x < 4, i.e. (−∞ ; 4)
2
Solve: 5x − 4 ≥ 16
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5x ≥ 20 → x ≥ 4, i.e. [4 ; ∞)
3
Solve: −2x > 10
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x < −5 (sign flips), i.e. (−∞ ; −5)
4
Solve: −3x + 1 ≤ 7
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−3x ≤ 6 → x ≥ −2, i.e. [−2 ; ∞)
5
Write x > 5 in interval notation.
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(5 ; ∞)
6
Write −2 ≤ x ≤ 7 in interval notation.
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[−2 ; 7]
7
Write (−∞ ; 3] as an inequality.
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x ≤ 3
8
Solve: −4 < 2x − 2 ≤ 6
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−1 < x ≤ 4 → (−1 ; 4]
9
On a number line, what does an OPEN circle mean?
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The value is not included (< or >).