Linear Inequalities & Interval Notation — Grade 10 (CAPS)

Solve linear inequalities and write the answer in interval notation. Covers number lines, compound inequalities and the rule that catches everybody: dividing by a negative flips the sign.

Also known as: solving inequalities · number line inequalities · 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.

The answer to an equation is usually one number. The answer to an inequality is a whole interval: a stretch of the number line, such as every number bigger than 2. Interval notation is a short way to write that stretch using its two endpoints and a bracket at each end that says whether the endpoint belongs to the answer.

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.

💡

Read an interval back as an inequality to check it: (−2 ; 5] means −2 < x ≤ 5. The left number always comes first, so (5 ; −2] is never correct.

4Variables on both sides

Worked ExampleSolve: 5x − 7 ≤ 2x + 8, and write the answer in interval notation

1
Subtract 2x from both sides: 3x − 7 ≤ 8.
Collect the x-terms on the side with the larger coefficient, so the coefficient stays positive.
2
Add 7 to both sides: 3x ≤ 15.
3
Divide by 3: x ≤ 5.
3 is positive, so the sign stays.
4
Interval notation: (−∞ ; 5].
Square at 5 because 5 is included.

5Brackets and a negative coefficient

Worked ExampleSolve: 3 − 2(x + 1) < 7

1
Expand: 3 − 2x − 2 < 7, so 1 − 2x < 7.
The −2 multiplies both terms in the bracket.
2
Subtract 1: −2x < 6.
3
Divide by −2 and flip the sign: x > −3.
Dividing by a negative reverses < into >.
4
Interval notation: (−3 ; ∞).
Round at −3 because −3 itself is not a solution.

6An inequality with a fraction

Worked ExampleSolve: (x − 1)/3 ≥ 2

1
Multiply both sides by 3: x − 1 ≥ 6.
3 is positive, so the sign stays.
2
Add 1: x ≥ 7.
3
Interval notation: [7 ; ∞).

7Compound 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).

Worked ExampleSolve: −3 ≤ 1 − 2x < 7

1
Subtract 1 from all three parts: −4 ≤ −2x < 6.
2
Divide all three by −2 and flip BOTH signs: 2 ≥ x > −3.
Every sign reverses when you divide by a negative.
3
Rewrite with the smaller number first: −3 < x ≤ 2.
The same statement, read from left to right.
4
Interval notation: (−3 ; 2].

8When 𝑥 must be an integer

Exam questions often add a condition such as x ∈ ℤ (x is an integer). Then the answer is not an interval but a list of the integers inside it.

Worked ExampleIf −1 < x ≤ 4 and x ∈ ℤ, list the values of x

1
−1 is excluded, so start at 0; 4 is included.
Read the two signs before you list anything.
2
x ∈ {0 ; 1 ; 2 ; 3 ; 4}.
Five integers, written in set brackets.

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 >).