Overview
What you’ll learn
- Read and write statements with the symbols <, >, ≤ and ≥.
- Represent inequality solutions on a number line with open and closed circles.
- Solve linear inequalities in one variable, including cases that need the sign flip.
- List the integer solutions that satisfy an inequality or a double-ended range.
- Form and solve linear inequalities from real-world problems.
In 30 seconds
- 1x > 5 means strictly bigger; x ≥ 5 means bigger or equal — “at least”.
- 2A number line uses an open circle for < or > and a closed circle for ≤ or ≥, shaded towards the solutions.
- 3Inequalities solve like equations: add, subtract, multiply and divide both sides to isolate x.
- 4Multiplying or dividing by a negative number flips the inequality sign — the one big difference from equations.
- 5A range such as 2 ≤ x < 7 means x is between the ends: including 2, excluding 7.
- 6Word problems hide the symbol: “at most” is ≤, “at least” is ≥, “more than” is >.
Notes · 1 of 6
The four inequality symbols
Two strict symbols and two “or equal” symbols carry all the meaning you need.
| Symbol | Read as | Example | Number-line circle |
|---|---|---|---|
| < | less than | x < 3 | open at 3, shade left |
| > | greater than | x > 3 | open at 3, shade right |
| ≤ | less than or equal to | x ≤ 3 | closed at 3, shade left |
| ≥ | greater than or equal to | x ≥ 3 | closed at 3, shade right |
The little bar underneath ≤ and ≥ is the “or equal to” part. Phrases map onto symbols: “at least 5” means x ≥ 5, “no more than 5” means x ≤ 5, “more than 5” means x > 5, and “below 5” means x < 5. The words at least and at most always bring the bar with them.
Notes · 2 of 6
Solving inequalities
Whatever you do to one side, do to the other — the equation habit carries over.
An inequality is solved exactly like an equation: undo the operations one at a time. From x + 4 > 9, subtract 4: x > 5. From 3x ≤ 12, divide by 3: x ≤ 4. The solution is usually a whole family of numbers, not a single value, which is why the answer keeps its sign.
| Inequality | Move | Solution |
|---|---|---|
| x − 7 > −2 | add 7 | x > 5 |
| 4x ≤ 20 | divide by 4 | x ≤ 5 |
| x/3 < 2 | multiply by 3 | x < 6 |
Notes · 3 of 6
The sign flip
Multiply or divide by a negative and the inequality turns around.
Here is the one rule equations never asked of you: multiplying or dividing both sides by a negative number reverses the sign. It has to — negating swaps which side is bigger, since 2 < 5 but −2 > −5. So from −2x < 8, dividing by −2 gives x > −4, not x < −4.
- 11. Move + terms first
Add or subtract to gather x-terms; these moves never flip anything.
- 22. Spot the negative coefficient
A leading minus on x means a flip is coming.
- 33. Divide and flip
Divide both sides by the negative number and reverse the sign in the same breath.
- 44. Sanity-check
Test one value from your answer in the original inequality.
Notes · 4 of 6
Integer solutions in a range
Many questions do not want x itself — they want which whole numbers work.
After solving, the boundary tells you which integers qualify. For x < 4 the integers are everything up to 3; for x ≤ 4 the integer 4 joins them. Watch the dot: the difference between < and ≤ is exactly one integer at the edge.
| Range | Integers | How many |
|---|---|---|
| 2 ≤ x < 7 | 2, 3, 4, 5, 6 | 5 |
| −2 < x ≤ 3 | −1, 0, 1, 2, 3 | 5 |
| 1 < x < 5 | 2, 3, 4 | 3 |
Notes · 5 of 6
Two conditions at once
When x must satisfy two inequalities, only the overlap survives.
A problem can demand two things: x > 1 and x ≤ 6. Written as one statement that is 1 < x ≤ 6 — read it as “between 1 and 6, including 6”. On a number line, both circles are drawn and only the shared stretch is shaded.
A safe routine: solve each condition separately, sketch both on one number line, and read off the overlap. If the conditions contradict — say x > 5 and x ≤ 2 — there is no number satisfying both, and saying so is a complete answer.
Notes · 6 of 6
Inequalities in real problems
Budgets, limits and “at least” targets translate straight into symbols.
| Phrase | Symbol | Example |
|---|---|---|
| at most / no more than / up to | ≤ | the total is at most $300 |
| at least / no less than | ≥ | savings of at least $145 |
| more than / over | > | more than 12 guests |
| fewer than / under | < | under 20 minutes |
Build the inequality the way you would build an equation — define the letter, write each side — then solve. Finish by answering the original question: a solution x ≤ 30 might mean at most 30 chairs, and the answer “30” needs the whole sentence, not just the number.
Revise
Key terms
The words examiners expect you to use precisely.
- Inequality
- A statement comparing two expressions with <, >, ≤ or ≥ instead of equals.
- Strict inequality
- A comparison with < or >, where equality is not allowed.
- At least
- Greater than or equal to (≥); equality is allowed.
- At most
- Less than or equal to (≤); equality is allowed.
- Number line
- A line on which solutions are shown as shaded stretches with open or closed circles.
- Open circle
- A hollow dot on a number line marking an excluded end point (< or >).
Show 6 more key termsShow fewer key terms
- Closed circle
- A filled dot on a number line marking an included end point (≤ or ≥).
- Sign flip
- Reversing the inequality sign when both sides are multiplied or divided by a negative number.
- Integer solutions
- The whole numbers (positive, negative or zero) that satisfy an inequality.
- Range
- A set of values between two ends, written as 1 < x ≤ 4.
- Boundary value
- The end number of a solution set; testing it reveals whether equality is allowed.
- Overlap
- The values satisfying two conditions at once, found where their shadings meet.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Forgetting to flip the sign when dividing by a negative.
Negating reverses which side is bigger, so the comparison must reverse too.
Instead: −2x < 8 → x > −4. The flip happens exactly when the negative division happens.
Mistake: Flipping the sign when adding or subtracting a negative.
The flip belongs to multiplication and division only.
Instead: x − 5 > −2 is solved by adding 5: x > 3, with no flip.
Show 8 more mistakesShow fewer mistakes
Mistake: Drawing an open circle for x ≥ 4 on a number line.
The “or equal to” bar means 4 itself is a solution.
Instead: ≤ and ≥ always use closed (filled) circles; < and > use open circles.
Mistake: Including the open end when counting integers, e.g. counting 7 in 2 ≤ x < 7.
An open end excludes that number.
Instead: Write the integer list and strike out open ends before counting.
Mistake: Reporting x ≥ 15 as the answer to “how many weeks at least?”.
The question asks for a number of weeks, not an inequality.
Instead: Solve, then answer in context: the minimum is 15 weeks.
Mistake: Treating “more than” as ≥.
“More than” is strict; “at least” allows equality.
Instead: More than 12 → x > 12; at least 12 → x ≥ 12.
Mistake: Checking the solution only in the rearranged line.
A sign error on the way to the answer hides from its own equation.
Instead: Substitute a test value into the original inequality — one value inside and, for tough cases, one outside.
Mistake: Insisting on a solution when the two conditions contradict, e.g. x > 3 and x < 1.
No number can be both above 3 and below 1, so forcing an answer invents one.
Instead: Sketch both conditions; if the shadings never overlap, the answer is “no solution”.
Mistake: Claiming 2x < 3x “because 2 < 3”, for every x.
Multiplying an inequality by a negative reverses it: with x = −1, 2x = −2 and 3x = −3, and −2 < −3 is false.
Instead: The direction of 2x vs 3x depends on the sign of x; multiply by an unknown at your peril.
Mistake: Chaining symbols in mixed directions, e.g. writing a combined condition as 3 < x > 5.
A chained range must read left-to-right with one direction; mixed symbols describe no single range.
Instead: Write the overlap with the smaller number on the left, e.g. 3 < x < 5 or its ≤ variants.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Translate phrases into symbols
Which inequality represents “x is at least 5”?
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 59 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 2 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 2.
Mark scheme
PROChapter 2 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 1Coming soonWeighted Assessment 1 (WA1)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 2 about?
Chapter 2 covers linear inequalities in one variable: the four inequality symbols, solving with the same moves as equations, the sign flip when multiplying or dividing by a negative, number-line diagrams, integer solutions in ranges, and word problems with limits.
When do I flip the inequality sign?
Exactly when you multiply or divide both sides by a negative number. Adding or subtracting anything — even a negative — never flips the sign.
What is the difference between an open and a closed circle?
An open circle marks an excluded boundary (for < and >); a closed circle marks an included boundary (for ≤ and ≥). “At least” and “at most” always take closed circles.
How do I count the integers in a range like 2 ≤ x < 7?
List them: 2, 3, 4, 5, 6 — five integers. The closed end includes 2 and the open end excludes 7, so subtracting the end numbers would miscount by one.
Up next
Chapter 3: Quadratic Expressions and Equations