ALGEBRA — Linear inequalities

Subject: Mathematics • Target age: 12 (Kenyan learners)

Learning outcomes

Understand what an inequality is and common inequality symbols (>, <, ≥, ≤).
Solve simple linear inequalities in one variable (ax + b < c, etc.).
Represent solutions on a number line (open and closed points).
Handle inequalities that require multiplication or division by a negative number.

1. What is an inequality?

An inequality compares two expressions and shows that one is greater or smaller than the other. Common symbols:

> means "greater than"    e.g. 5 > 3
< means "less than"    e.g. 2 < 4
≥ means "greater than or equal to" e.g. 6 ≥ 6
≤ means "less than or equal to"    e.g. 1 ≤ 3

2. Solving linear inequalities — steps

Use the same rules as solving equations: add or subtract to move terms, then multiply or divide to isolate the variable.
Important rule: if you multiply or divide both sides by a negative number, you must reverse (flip) the inequality sign.
Check your answer by picking a number from the solution and substituting it back into the original inequality.

3. Examples with steps

Example 1

Solve 2x + 3 < 9

Step 1: Subtract 3 from both sides → 2x < 6
Step 2: Divide both sides by 2 → x < 3

Solution set: { x | x < 3 } which means any number less than 3.

-3 -2 -1 0 1 2 3

Number line: open circle at 3 and shade left — all numbers less than 3.

Example 2 (important rule)

Solve -3x ≥ 9

Step 1: Divide both sides by -3. Because we divide by a negative number, flip the inequality sign. → x ≤ 9 ÷ (-3) = -3

Solution: x ≤ -3 (all numbers that are -3 or less).

-5 -4 -3 -2 -1

Number line: closed circle at -3 and shade left — x ≤ -3.

4. Compound inequalities

Example: 1 ≤ x + 2 < 5 Step 1: Subtract 2 from all parts → 1 − 2 ≤ x < 5 − 2 → −1 ≤ x < 3. This means x is at least −1 and less than 3. On a number line, put a closed circle at −1 and an open circle at 3, shade between them.

5. Quick tips

Do the same steps you use for equations, but remember to flip the sign if you multiply or divide by a negative number.
Open circle = not included ( < or > ). Closed circle = included ( ≤ or ≥ ).
Always check by substituting one value from your solution set into the original inequality.

6. Practice exercises

Solve: 3x − 4 > 5
Solve: −2x ≤ 6
Solve and show on a number line: x + 5 < 2
Solve the compound inequality: −2 < 2x + 1 ≤ 5
True or false? If x < 4, then 2x < 8.

Answers

3x − 4 > 5 → 3x > 9 → x > 3
−2x ≤ 6 → divide by −2 (flip sign) → x ≥ −3
x + 5 < 2 → x < −3 (open circle at −3, shade left)
−2 < 2x + 1 ≤ 5 → subtract 1: −3 < 2x ≤ 4 → divide by 2: −1.5 < x ≤ 2
True. If x < 4 then multiplying both sides by 2 gives 2x < 8.

7. Key vocabulary

Inequality, less than, greater than, less than or equal to, greater than or equal to, solution set, number line, open circle, closed circle.

Study note: Practice many questions. Use number lines to check your answers — they show very clearly whether points are included or not. Good work!

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Topics

Linear inequalities Notes, Quizzes & Revision