Last reviewed: 7 November 2026
Updated November 2026. These Grade 9 linear inequalities notes show how to solve inequalities in one unknown, show the answer on a number line, and represent inequalities in one and two unknowns on a graph. Solving an inequality is like solving an equation, with one extra rule: when you multiply or divide both sides by a negative number, reverse the inequality sign. For example, −2x ≥ 6 gives x ≤ −3.
The examples and questions are original practice for Kenyan junior school learners, not an official paper. For the other Grade 9 maths topics, see our Grade 9 maths revision guide and the Grade 9 notes hub. If equations feel shaky, revise linear equations for junior school first.
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What the Grade 9 Curriculum Covers
Linear inequalities is a sub-strand under Algebra in the Kenya Institute of Curriculum Development’s Grade 9 Mathematics design. Learners solve linear inequalities in one unknown, represent them graphically, represent linear inequalities in two unknowns graphically, and apply them to real life. The design is listed on the KICD Grade Nine curriculum designs page.
Inequality Signs
| Sign | Meaning | Example |
|---|---|---|
| < | less than | x < 4 |
| > | greater than | x > −1 |
| ≤ | less than or equal to | x ≤ 7 |
| ≥ | greater than or equal to | x ≥ 0 |
Real-life words help: “at most 40 learners” means x ≤ 40, and “more than KSh 500” means x > 500.
Solving Inequalities in One Unknown
Use the same steps as for an equation: add or subtract the same amount on both sides, then multiply or divide. The method follows OpenStax Elementary Algebra’s section on solving linear inequalities.
Worked example one
Solve 3x − 5 < 10.
- Add 5 to both sides: 3x < 15.
- Divide by 3: x < 5.
Worked example two: dividing by a negative
Solve 4 − 2x ≥ 10.
- Subtract 4: −2x ≥ 6.
- Divide by −2 and reverse the sign: x ≤ −3.
Check with a value: x = −4 gives 4 − 2(−4) = 12, and 12 ≥ 10 is true. x = 0 gives 4, which is not ≥ 10. So the answer x ≤ −3 is correct.
Worked example three: a compound inequality
Solve −1 < 2x + 3 ≤ 9. Subtract 3 from all three parts: −4 < 2x ≤ 6. Divide by 2: −2 < x ≤ 3. The integer solutions are −1, 0, 1, 2 and 3.
Showing Inequalities on a Number Line
- Use an open circle for < or > (the number is not included).
- Use a filled circle for ≤ or ≥ (the number is included).
- Draw an arrow or line over the values that satisfy the inequality.
For −2 < x ≤ 3: an open circle at −2, a filled circle at 3, and a line joining them.
Inequalities in Two Unknowns on a Graph
To show x + y ≤ 4 on a Cartesian plane:
- Draw the boundary line x + y = 4 using a table of values, for example (0, 4) and (4, 0).
- Use a solid line for ≤ or ≥, and a broken (dashed) line for < or >.
- Test a point not on the line, such as (0, 0): 0 + 0 = 0 ≤ 4 is true, so (0, 0) is on the wanted side.
- Shade to show the region. Many Kenyan teachers shade the unwanted region and leave the wanted region clear; some books do the opposite. Follow your teacher’s convention and say which one you used.
The same steps work for y > 2x − 1: dashed line through (0, −1) and (1, 1), and test (0, 0): 0 > −1 is true, so the wanted region contains the origin.
Real-life example
In this made-up example, a learner has KSh 200 to buy pens at KSh 20 (x) and books at KSh 50 (y). The spending condition is 20x + 50y ≤ 200, with x ≥ 0 and y ≥ 0. Any whole-number point in the wanted region, such as (5, 2), is an affordable combination: 100 + 100 = 200.
Original Practice Questions With Worked Answers
- Solve 5x + 2 > 17. 5x > 15, so x > 3.
- Solve 7 − 3x ≤ 1. −3x ≤ −6, so x ≥ 2 (sign reversed).
- List the integers that satisfy −3 ≤ x < 2. −3, −2, −1, 0, 1.
- Is the boundary of y < x + 2 solid or broken? Broken, because the sign is strict (<).
- Does (1, 1) satisfy 2x + y ≥ 4? 2 + 1 = 3, which is not ≥ 4. No.
Cover the answers and attempt each question first. For timed practice, use the Grade 9 revision papers or the Grade 9 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
When do you reverse the inequality sign?
Whenever you multiply or divide both sides of an inequality by a negative number.
What is the difference between an open and a closed circle on a number line?
An open circle means the number is not included (< or >). A closed circle means it is included (≤ or ≥).
How do I know which side of the line to shade?
Test a point that is not on the line, such as (0, 0). If it satisfies the inequality, it is in the wanted region.
When is the boundary line dashed?
When the inequality is strict (< or >), because points on the line are not included.