Secondary 1 G3 Mathematics Tutor Bukit Timah | Solving Simple Inequalities and Negative Coefficients

Did you know? Dividing an inequality by a negative number reverses its direction. That’s why −2x < 8 gives x > −4, not x < −4. Many Secondary 1 learners can solve 2x = 8 confidently but forget that an inequality describes a whole set of possible values, rather than one answer.

If you’re helping a Secondary 1 G3 Mathematics student in Bukit Timah, this is a practical bridge from comparing numbers to solving simple linear inequalities. We’ll start with what the symbols mean, then solve and check examples, including negative coefficients. Every result is tested on a number line so the sign change has an explanation.

Quick answer: how to solve a simple inequality

To solve 3x ≤ 12, divide both sides by positive 3. The direction stays the same, giving x ≤ 4. That means every real number no greater than 4 satisfies the inequality.

To solve −3x ≤ 12, divide both sides by negative 3. Reverse the inequality sign to obtain x ≥ −4. Check two values: x = 0 makes 0 ≤ 12 true, while x = −5 makes 15 ≤ 12 false. That agrees with the set x ≥ −4.

Three habits make this reliable: preserve the direction when adding or subtracting the same amount on both sides, preserve it when multiplying or dividing by a positive number, and reverse it when multiplying or dividing by a negative number.

What is an inequality?

An equation such as x + 2 = 5 says two expressions are equal; its solution is x = 3.

An inequality such as x + 2 < 5 says the left side must be smaller than the right. Subtracting 2 from both sides gives x < 3. Values such as x = 0, x = 2 and x = −5 work, but x = 3 does not.

The four comparison symbols are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). Read the symbols aloud before calculating.

If the student still confuses these, begin with the inequality-symbol and negative-number-line guide. Solving is easier when the basic relationship already makes sense.

Why a negative multiplier flips the inequality

Start with a true statement: 2 < 5. Multiply both sides by 2. We get 4 < 10, still true.

Now multiply both sides by −1. The results are −2 and −5. On the number line, −2 lies to the right of −5, so the true comparison is −2 > −5. The order has reversed.

The same happens when dividing by a negative number. From −2x < 8, dividing by −2 gives x > −4.

The rule isn’t arbitrary. A negative scaling reverses the order of real numbers around zero. If a tutor demonstrates that with a number line before showing the symbolic shortcut, students are more likely to remember why the direction changes.

Worked example 1: a positive coefficient

Solve 5x > 20.

Divide both sides by positive 5: x > 4. The solution excludes 4 itself, because 5 × 4 = 20 is equal to, not greater than, 20.

Try x = 5. The original inequality becomes 25 > 20, which is true. Try x = 3. The original becomes 15 > 20, false.

On a number line, use an open circle at 4 and shade to the right. The open circle indicates the boundary is not included.

Worked example 2: a negative coefficient

Solve −4x ≥ 12.

Divide by −4 and reverse the sign: x ≤ −3.

Check x = −3. Then −4(−3) = 12, and 12 ≥ 12 is true, so the endpoint is included.

Check x = −4. The left side becomes 16 ≥ 12, true. Check x = −2. The left side becomes 8 ≥ 12, false.

The complete answer is x ≤ −3. On a number line, use a filled circle at −3 and shade left.

Worked example 3: one operation before dividing

Solve 3x + 5 < 17.

Subtract 5 from both sides: 3x < 12. Divide by positive 3: x < 4.

A common mistake is to divide 17 by 3 while leaving the +5 untouched. The coefficient multiplies x, but the extra 5 is a separate term that must be accounted for.

Check x = 3: 3(3) + 5 = 14 < 17, true. Check x = 4: 17 < 17, false. The strict inequality correctly excludes 4.

Worked example 4: a negative after subtraction

Solve 7 − 2x > 15.

Subtract 7: −2x > 8. Divide by negative 2 and reverse the comparison: x < −4.

Check x = −5: 7 − 2(−5) = 17 > 15, true. Check x = −3: 7 + 6 = 13 > 15, false.

Be careful not to write x > −4 after dividing by the negative coefficient. One reversed sign is the difference between an entirely correct and entirely incorrect set of solutions.

Worked example 5: fractions can still lead to simple inequalities

Solve x/3 ≤ −2.

Multiply both sides by positive 3. The direction stays the same, giving x ≤ −6.

Check x = −9. Then −9/3 = −3 ≤ −2, true. Check x = −3. Then −1 ≤ −2 is false.

If the inequality instead read −x/3 ≤ −2, multiplying both sides by −3 would reverse the symbol and yield x ≥ 6. The negative multiplier, not the presence of a fraction, is what changes the direction.

A second method: rearrange before dividing

Take −5x + 2 ≥ 17.

Subtract 2 from each side to obtain −5x ≥ 15. Dividing by −5 gives x ≤ −3.

You can avoid dividing by a negative coefficient by adding 5x to both sides instead. From −5x + 2 ≥ 17, add 5x: 2 ≥ 17 + 5x. Subtract 17: −15 ≥ 5x. Divide by positive 5: −3 ≥ x, which is the same as x ≤ −3.

Both methods are valid. The important thing is that each new statement has the same solution set as the previous one.

The boundary test: open or closed circle?

The solution x < 2 excludes x = 2. An open circle at 2 shows that exclusion, and shading extends left.

The solution x ≤ 2 includes x = 2. A filled circle shows inclusion, with shading left.

Similarly, x > 2 shades right with an open circle, while x ≥ 2 shades right with a filled one.

The arrow or ray describes all values allowed, not just a few listed integers. Between 1 and 2 are many decimals and fractions. If the question restricts x to integers, then list integer values accordingly. Read the permitted domain.

Translating an everyday condition into an inequality

Suppose a fictional event rule says that a group may have at most 12 participants. Let n be the number of participants. The statement is n ≤ 12.

Because n counts people, only whole numbers make sense; negative numbers and 4.5 people are not valid in that context.

Now suppose an ordinary ticket costs $3 per participant and the fictional total budget is at most $30. The inequality is 3n ≤ 30, so n ≤ 10. If n represents a nonnegative whole-number count, the permissible values are 0 through 10.

Do not invent an integer restriction for every inequality. The story may specify a continuous quantity such as length or time instead.

What makes an inequality answer wrong even when the algebra looks neat?

One student writes −3x < 6, then x < −2. The arithmetic division is correct, but the sign direction is not. The true solution is x > −2.

Another student solves 4x ≤ 20 and writes x = 5, forgetting all the smaller values. The boundary is 5, but the solution is x ≤ 5.

A third student solves x + 4 < 10 and draws a filled circle at 6. The number-line sketch includes a value that the original strict inequality excludes.

These are three different mistakes: reversing a comparison, confusing a boundary with the solution set, and misrepresenting endpoint inclusion.

Twelve-question mini diagnostic

• A. Solve 2x < 10.
• B. Solve −2x < 10.
• C. Solve 3x ≤ 12.
• D. Solve −3x ≤ 12.
• E. Solve 4x + 2 > 18.
• F. Solve 5 − 2x ≥ 11.
• G. Solve x/4 > 3.
• H. Solve −x/4 > 3.
• I. Is x = 6 allowed in x ≤ 6?
• J. Is x = 6 allowed in x < 6?
• K. Translate “at least 15” into a comparison.
• L. Translate “no more than 7” into a comparison.

Answers with one-line reasons

A. x < 5; divide by positive 2.

B. x > −5; dividing by negative 2 reverses the sign.

C. x ≤ 4.

D. x ≥ −4.

E. 4x > 16, so x > 4.

F. −2x ≥ 6, so x ≤ −3.

G. x > 12.

H. x < −12, since multiplication by −4 reverses the inequality.

I. Yes. The symbol ≤ includes equality.

J. No. The symbol < excludes equality.

K. x ≥ 15, for a quantity x.

L. x ≤ 7, for a quantity x.

Before marking, ask the learner to choose one allowed value and one disallowed value for each inequality. Substitution back into the original comparison makes the answer less dependent on memorised symbol-flipping.

A practical fifteen-minute revision

For the first three minutes, compare positive and negative numbers on a number line. Then solve two positive-coefficient inequalities and describe the shaded region.

Next practise two negative-coefficient inequalities, deliberately explaining the reversed direction. In the final five minutes, include a fixed term and one practical situation with whole-number restrictions.

A changed question a few days later is more informative than the same question repeated immediately. The student should be able to give a solution set, endpoint symbol and verbal explanation independently.

What Bukit Timah parents can ask a tutor

Ask whether the child can distinguish reading inequalities from solving them. Can the learner explain the change in direction after multiplying by a negative number? Do they check a boundary value in the original question?

The eduKateSG Secondary 1 Mathematics small-group tutorial reference describes focused three-student support near Sixth Avenue MRT. The Bukit Timah G3 Mathematics tuition overview provides local programme context. Please confirm lesson fit, current arrangements and school chapter order directly.

The MOE G2 and G3 Mathematics syllabus covers simple inequalities alongside equations in the lower-secondary G3 pathway. The examples here build on sign and equation understanding; more complicated inequality systems belong to later study.

Frequently asked questions

### Why does a negative coefficient reverse an inequality sign?

Multiplying or dividing both sides of a comparison by a negative number reverses the order of the real numbers. For instance, 2 < 5 becomes −2 > −5 after multiplication by −1.

### Should I write x = 4 when the answer is x ≤ 4?

No. x = 4 gives only one value; x ≤ 4 includes all values equal to or less than four.

### Is an open circle always used for < or >?

On a standard number-line solution diagram, an open circle indicates an excluded endpoint. A filled circle indicates an included endpoint.

### Can an inequality have negative solutions?

Yes. Check those values in the original inequality. A negative value is not automatically a mistake.

### How is an inequality different from an equation?

An equation states equality, while an inequality states an ordering. A simple inequality generally has a set of allowed values, not just a single solution.

Keep learning

• Inequality symbols and negative number lines
• Solving multi-step linear equations
• Negative number sign errors
• Translating words into algebraic expressions
• Ratio word problems using the unitary method

When a student solves an inequality, the finishing question is not “What number did I get?” It is “Which values really make this statement true?”