Grade 7 Mathematics ALGEBRA – Linear inequalities Notes
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
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!