IP EMaths Notes (Upper Sec, Year 3-4): 04) Linear Equations and Inequalities

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Isolate variables cleanly and represent inequality solutions in interval or graphical form.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: linear equations, fractional equations reducible to linear form, inequalities, number-line representation, and contextual interpretation form the main route.
School-sensitive extension: compound or multi-condition inequality systems and proof-style justification may vary by school.
2027 national comparison: K310 Topic N7 includes linear equations, simple fractional equations, linear inequalities in one variable, and number-line solutions.
Check your school: confirm interval-notation expectations and whether compound inequalities are assessed.
Exam-track route: use the separate O-Level and SEC G3 Mathematics notes for K310 topic ownership.

Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 04) Linear Equations and Inequalities cover?
A: Isolate variables cleanly and represent inequality solutions in interval or graphical form.

The core idea is simple: Do the same operation to every side, and flip an inequality when multiplying or dividing by a negative.

Use it as a working check: Linear questions are mostly bookkeeping. Clear fractions, collect like terms, isolate the variable, then show the final answer as a value, interval, or shaded number line.

Then go one layer deeper: Use the examples to see how equations and inequalities differ: equations pin down exact values, while inequalities describe ranges that may or may not overlap.

Linear manipulations appear everywhere - simultaneous systems, variation models, and coordinate geometry. Practice consistent isolation and be fluent with inequality notation.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

Key reminders

Keep the equation balanced: whatever operation you apply to one side must be applied to the other.
When multiplying or dividing an inequality by a negative number, reverse the inequality symbol.

Reviewed by

Marcus Pang·Managing Director (Maths)

Question cue	First move	Output form	Common trap
Contains $=$ only	Clear brackets or fractions, collect like terms, then isolate the variable.	Exact value such as $x = 5$ .	Treating the answer as a range.
Contains an inequality symbol	Solve like an equation, but track whether you divide by a negative number.	Inequality or interval notation.	Forgetting to reverse the sign after dividing by a negative.
Compound inequality with one expression between two bounds	Apply each operation to every part of the inequality.	One continuous interval if both sides overlap.	Working on only the left or right inequality.
Two separate inequalities joined by "and"	Solve both separately, then take the intersection.	Shared region on a number line.	Taking the union when the question requires both conditions.
Inequality plus integer or multiple condition	Solve the range first, then filter the permitted values.	Listed values, or "no solution" if none remain.	Keeping an endpoint that is excluded by a strict inequality.

Operation applied to every side	Flip the sign?	Why	Common trap
Add or subtract the same expression	No	The order of the two sides is preserved.	Flipping just because a negative term appears in the line.
Multiply or divide by a positive number	No	Positive scaling keeps the order the same.	Forgetting that division by $2$ , $0.5$ , or another positive value keeps the symbol direction.
Multiply or divide by a negative number	Yes	Negative scaling reverses the order on the number line.	Solving $-3x < 12$ as $x < -4$ instead of $x > -4$ .
Simplify first, then divide by the coefficient	Depends on the coefficient	Decide at the final division step.	Seeing a negative bracket earlier and flipping before the actual division step.

IP EMaths Notes (Upper Sec, Year 3-4): 04) Linear Equations and Inequalities

How this chapter applies

Key reminders

Sources

Equation-or-inequality checkpoint

Inequality sign-flip checkpoint

Worked example - Solve and interpret

Worked example - Solve a linear equation with fractions

Worked example - Solve a compound inequality

Worked example - Intersection of two independent inequalities

Worked example - Three-condition inequality system

Practice Quiz

Try this

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