Secondary 4 G2 Mathematics Tutor Bukit Timah | Inequalities, Number Lines and Real-World Constraints. Did you know? Multiplying both sides of an equation by −2 keeps the equal sign unchanged. Yet multiplying both sides of an inequality by −2 reverses the inequality sign. One tiny symbol carries an important idea about how negative numbers change order.
In Secondary 4 G2 Mathematics, inequalities are not merely equations with a different sign. They describe a range of possible answers. A student might be choosing how many notebooks fit a budget, interpreting a measurement limit or explaining why a value is not permitted. We want students in Bukit Timah to understand the boundary before they begin calculating.
The official 2027 SEC G2 Mathematics syllabus K210 includes number-line ordering and symbols such as <, >, ≤ and ≥ under N1. Under N7 it specifies solving simple inequalities of the form ax < b or ax > b, where a and b are integers. The mathematics studied for a 2026 N(A)-Level candidate must follow the applicable 4045 syllabus instead. The SEAB 2027 subject directory confirms the new K210 code; use the right examination-year document before selecting revision questions.
Quick answer: how to solve a simple inequality without guessing
- Read the inequality aloud. For 3x < 12, say “three times x is less than twelve.”
- Use the same legal operations on both sides. Add or subtract a quantity as with an equation.
- Check the sign of a divisor. When dividing or multiplying by a negative quantity, reverse < to > and vice versa.
- Interpret the boundary. Does the answer include equality, or must the value be strictly smaller or larger?
- Test sample values. Substitute a value inside the proposed region and one outside it.
An inequality is a statement of order. Since −6 is less than −2, after dividing by −2 the order becomes 3 greater than 1. This explains the direction change rather than relying on a memorised rule with no meaning.
The four comparison symbols and their number-line meaning
| Symbol | Meaning | How to show a boundary |
|---|---|---|
| x < 4 | x is strictly smaller than 4 | Open circle at 4; shade left. |
| x ≤ 4 | x is 4 or smaller | Filled circle at 4; shade left. |
| x > 4 | x is strictly greater than 4 | Open circle at 4; shade right. |
| x ≥ 4 | x is 4 or greater | Filled circle at 4; shade right. |
The number line is an aid to reasoning: the filled circle tells the reader that the boundary itself works. An open circle means it does not. The syllabus distinguishes number-line language under N1 from the specifically listed N7 simple linear-inequality forms, so follow the actual wording of the student’s question.
Eighteen original worked inequalities and boundary checks
1. A positive divisor
Solve 3x < 12. Divide both sides by 3: x < 4. Test x = 3: 9 < 12, true; x = 5: 15 < 12, false.
2. Greater than with a positive coefficient
Solve 5x > 20. Divide by positive 5: x > 4. The direction stays the same.
3. Divide by a negative coefficient
Solve −2x < 8. Divide by −2 and reverse the sign: x > −4. Test x = 0: 0 < 8, true.
4. Another negative-coefficient example
Solve −3x > 12. Divide by −3 and reverse: x < −4. Test x = −5: 15 > 12.
5. Negative number on the right
Solve 4x < −20. Divide by 4: x < −5. Check x = −6 gives −24 < −20.
6. Two negatives together
Solve −5x > −15. Divide by −5 and reverse: x < 3. Check x = 0: 0 > −15.
7. A non-integer boundary
Solve 4x < 11. Divide by 4: x < 11/4, or x < 2.75. The problem has real-number solutions, not only whole numbers.
8. A strict lower boundary
Solve 2x > −7. Divide by 2: x > −3.5. On the number line, shade right from an open circle at −3.5.
9. Test the boundary itself
For x < 4, the value x = 4 is false because 4 is not less than 4. The boundary is excluded.
10. A non-strict condition in everyday language
A room holds at most 24 people. If n counts people, n ≤ 24. Here n is a nonnegative whole number; an expression such as 24.5 people is not meaningful.
11. An upper price limit
A student can spend less than $30 on a purchase of one item priced p dollars. The limit is p < 30, not p ≤ 30. The phrase “less than” excludes equality.
12. A minimum requirement
A club requires at least 12 completed tasks. If n is the integer task count, n ≥ 12. Equality is allowed.
13. Comparing negative temperatures
Which values are greater: −7 or −2? The answer is −2 > −7, because −2 lies to the right of −7 on a number line.
14. Transform both sides by a negative factor
Starting from 2 < 6, multiply both sides by −3: −6 is greater than −18. Correct statement: −6 > −18.
15. Number-line interpretation of a strict condition
The condition x > 1.5 has an open circle at 1.5 and shading to the right. Test x = 2 works and x = 1 does not.
16. Number-line interpretation of an inclusive condition
The condition x ≤ −3 includes −3 itself and everything smaller. A filled circle at −3 and shading left give the correct representation.
17. An integer-only budget model
A notebook costs $4. A budget of less than $19 allows positive integer quantity n satisfying 4n < 19. Since n < 4.75 and n is an integer, the greatest affordable quantity is 4 notebooks.
18. Comparing possible solutions with restrictions
Suppose x < 6 and x represents a nonnegative whole number. Permissible values are 0, 1, 2, 3, 4, 5. If x represents a positive number of students instead, 0 is excluded. Context determines the set.
Why reversing the inequality is mathematically necessary
Start with 1 < 3. Subtracting the same number from each side keeps 1 − 5 < 3 − 5: −4 < −2. Now multiply the original comparison by −2. The resulting numbers are −2 and −6. Since −2 lies to the right of −6, the correct statement becomes −2 > −6. Reversing is a consequence of order on the number line.
In a helpful tutorial, students explain the rule using actual numbers before applying it to −5x < 20. This one contrast catches an entire class of repeated mistakes.
What the first wrong step reveals
| Incorrect attempt | What’s really wrong | Targeted retest |
|---|---|---|
| −2x < 8 becomes x < −4 | Sign not reversed after division by −2 | Test x = 0 against the original statement. |
| x < 5 shown with a filled boundary | Strict condition was mistaken for inclusive | Does x = 5 satisfy 5 < 5? |
| For ‘at most 8’ writes n < 8 | The word ‘at most’ was misread | Is n = 8 permitted by the stated rule? |
| For 3x > 12 writes x < 4 | Sign reversed after positive division | Which operation was applied, and was it negative? |
| Gives x = 4 as the sole solution to x > 4 | A region was mistaken for a single value | Can 5, 6, or 100 also satisfy the condition? |
| Reports 4.75 notebooks | Real-number solution not reconciled with whole objects | Is an incomplete notebook permitted in the model? |
Five no-hint questions to test mastery
- Solve −4x < 20.
- Solve 7x > −21.
- Represent x ≤ 2 on a number line.
- A teacher needs at least 18 sheets of paper. Write a suitable inequality for whole-number sheet count n.
- A ticket costs $7 and a pupil may spend at most $40. What is the largest whole number of tickets possible?
Answers: (1) x > −5; (2) x > −3; (3) a filled circle at 2, shaded left; (4) n ≥ 18; (5) 7n ≤ 40 means n ≤ 40/7, so the maximum whole number is 5. Notice which answers describe real numbers and which require integer interpretation.
What could happen in a three-student, 90-minute tutorial?
| Minutes | Activity | Independent evidence |
|---|---|---|
| 0–10 | Explain ordering of negative numbers without notes | Student knows why −3 is larger than −8. |
| 10–25 | Diagnostic with six simple ax < b or ax > b tasks | Tutor identifies the first wrong sign decision. |
| 25–50 | Contrast positive and negative operations on both sides | Learner justifies any reversed symbol. |
| 50–75 | Individually solve changed number-line and budget questions | Student distinguishes strict and inclusive bounds. |
| 75–90 | Exit question, error note, delayed retest | Transfer checked on a different inequality later. |
The immutable eduKateSG Secondary 1 Mathematics small-group tutorial reference describes premium three-pupil lessons near Sixth Avenue MRT, typically 1.5 hours weekly. The teaching sequence above is an illustrative way to apply that format to G2 inequalities; it does not confirm the current availability or pricing of a specific Secondary 4 G2 class.
A one-week repair plan
- Monday: put negative and positive values into number-line order.
- Tuesday: practise six inequalities with positive coefficients.
- Wednesday: practise six with negative coefficients; justify every reversal.
- Thursday: distinguish <, >, ≤ and ≥ using sample values.
- Friday: translate at most, at least and strictly less into mathematical notation.
- Saturday: solve a mixed set without clues about method or sign.
- Sunday: revisit mistakes after a delay with changed numbers.
Frequently asked questions
Is every complicated inequality tested at G2?
No. The published 2027 K210 N7 entry specifies simple forms ax < b and ax > b with integer a and b. Number ordering and other comparison symbols are also part of N1. This guide uses straightforward practical constraints to strengthen understanding, not to claim that compound inequalities or advanced algebraic inequalities are separate required topics.
Why does the sign reverse when dividing by a negative?
Negative multiplication reverses order on the number line. For example, −6 is less than −2, but dividing by −2 produces 3 greater than 1. A few numerical checks make the rule memorable and testable.
Should the answer always be an integer?
No. The algebraic inequality can describe real numbers. If the question refers to whole objects or people, the final real-world interpretation may restrict the answer to nonnegative or positive integers.
How can parents tell whether tutoring is helping?
Ask the learner to explain an unfamiliar negative-coefficient inequality without notes and show the solutions on a number line. Recheck the method several days later, not only immediately after a tutor demonstration.
Continue the Secondary 4 G2 Mathematics series
- Equations and simultaneous solutions
- Number sense and indices
- Real-world mathematical modelling
- Paper 1 and Paper 2 exam strategy
Official sources: SEAB 2026 N(A) syllabus list · SEAB 2027 SEC G2 syllabus list · K210 Mathematics syllabus N1 and N7. For tuition information see the unchanged eduKateSG three-student Mathematics tutorial page or eduKate Singapore. Examples are original practice questions, not official SEAB papers.