Sec 1 Math Linear Equations & Simple Inequalities Notes

Q: How do I turn a word problem into an equation?

Name the unknown first (“let y be…”), translate each phrase into algebra, then build one equation from the sentence that states a total or a comparison. Solve, and answer the exact question asked with units.

Q: When is this chapter tested?

Linear equations and inequalities usually appear in Weighted Assessment 2 and the end-of-year exam. Every school sets its own scope, so check your school’s assessment outline.

Overview

What you’ll learn

Solve linear equations in one variable using inverse operations, and check solutions by substitution.
Solve equations with brackets and with fractional coefficients.
Solve simple fractional equations that reduce to linear equations.
Form linear equations from word problems, solve them, and interpret the answers in context.
Solve simple linear inequalities and represent their solutions on a number line, flipping the sign when dividing by a negative.

In 30 seconds

1An equation is a balance: do the same thing to both sides and it stays true.
2Undo operations in reverse order: last applied, first undone — brackets and fractions clear early.
3Check every solution by substituting it back; a wrong sign usually shows up immediately.
4Word problems: name the unknown, build the equation, solve, then answer the actual question with units.
5Inequalities behave like equations — except dividing or multiplying by a negative flips the sign.
6On a number line, a closed dot (●) means the endpoint is included; an open dot (○) means it is not.

Notes · 1 of 5

Solving linear equations

Peel the equation back to x = something, doing the same thing to both sides every step.

A linear equation has the unknown to the power 1 only, like 3x − 4 = 11. Think of the equals sign as a balance scale: adding 4 to both sides keeps it level. The goal is always the same — get x alone.

1Clear fractions and brackets first
Multiply both sides by the denominator; expand brackets.
2Move number terms
Add or subtract to gather numbers on one side, x-terms on the other.
3Divide by the coefficient
3x = 15 gives x = 5.
4Check
Substitute back: 3 × 5 − 4 = 11 ✓.

Notes · 2 of 5

Brackets and fractional equations

Clear what wraps the x first — or divide the whole equation to shrink it.

Two standard openings
Equation	First move	Then
2(x + 3) = 14	Expand: 2x + 6 = 14	2x = 8, x = 4
2(x + 3) = 14	Or divide by 2: x + 3 = 7	x = 4 (faster)
(x + 5)/2 = 7	Multiply by 2: x + 5 = 14	x = 9
(2x − 1)/3 = 5	Multiply by 3: 2x − 1 = 15	2x = 16, x = 8

With the fraction bar, the line acts as brackets around the whole top: multiplying both sides by the denominator clears it in one step. Equations like (2x − 1)/3 = 5 are exactly why: after clearing, they are ordinary two-step equations.

Notes · 3 of 5

Forming equations from words

Name the unknown, translate sentence by sentence, solve, and answer the real question.

Translating phrases
Words	Algebra
5 more than a number n	n + 5
twice a number, decreased by 3	2n − 3
Wei has $12 more than Jun (Jun has $y)	y + 12
the length is 3 cm shorter than twice the breadth	2b − 3

1Define the unknown
Write it down: “Let y be the amount Jun has (in dollars).”
2Translate each sentence
Turn every phrase into algebra as you read.
3Solve and check
The solution must make sense in the story (no negative ages).
4Answer the question asked
If it asks for Wei’s money, do not stop at Jun’s.

Notes · 4 of 5

Simple inequalities

One rule changes everything: multiplying or dividing by a negative flips the sign.

The four symbols
Symbol	Meaning	Example solution
<	less than	x < 4: values left of 4, open dot
>	greater than	x > −1: values right of −1, open dot
≤	less than or equal to	x ≤ 4: includes 4 itself, closed dot
≥	greater than or equal to	x ≥ −1: includes −1 itself, closed dot

Solving works like equations — same operation on both sides — with one exception. Because the number line reverses when you scale by a negative: −2x > 10 becomes x < −5, not x > −5. Test it: x = −6 satisfies −2(−6) = 12 > 10 ✓, and −6 < −5 ✓; x = −4 fails (8 < 10).

Adding or subtracting anything: sign keeps its direction.
Multiplying or dividing by a positive: sign keeps its direction.
Multiplying or dividing by a negative: flip the sign.

Notes · 5 of 5

Showing solutions on the number line

An inequality solution is a whole ray of numbers — draw it, endpoint behaviour and all.

Drawing the four answer types
Solution	Endpoint	Direction
x < 4	open dot at 4 (excluded)	shade left
x ≤ 4	closed dot at 4 (included)	shade left
x > -1	open dot at -1 (excluded)	shade right
x ≥ -1	closed dot at -1 (included)	shade right

The dot carries the “or equal to”: closed when the endpoint satisfies the inequality, open when it does not. The shading shows every value that works — which is why inequality answers are sentences (“x ≤ 4”), not single numbers like equation answers.

Revise

Key terms

The words examiners expect you to use precisely.

Linear equation: An equation where the unknown appears to the power 1 only, e.g. 3x − 4 = 11.
Solution: The value of the unknown that makes the equation true.
Inverse operation: The operation that undoes another: addition undoes subtraction, division undoes multiplication.
LHS / RHS: The left-hand side and right-hand side of the equals sign; solving keeps them equal.
Fractional equation: An equation with the unknown in a fraction, e.g. (2x − 1)/3 = 5, solved by multiplying both sides by the denominator.
Inequality: A statement comparing two expressions with <, >, ≤ or ≥, with many possible solutions.

Show 4 more key terms

Flip the inequality: Reverse the direction of < or > when multiplying or dividing both sides by a negative number.
Closed dot: A filled point on a number line showing the endpoint is included (≤ or ≥).
Open dot: A hollow point on a number line showing the endpoint is excluded (< or >).
Check by substitution: Replacing the unknown with your solution to confirm both sides match.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Doing an operation to one side only, e.g. solving 3x − 4 = 11 as 3x = 11.
The balance breaks: the equation is no longer true after changing just one side.
Instead: Whatever you do to one side, do to the other: 3x = 15, then x = 5.
Mistake: Clearing a fraction by multiplying only the top: (x + 5)/2 = 7 becomes x + 5 = 7.
The whole left side is divided by 2, so both sides must be multiplied by 2 — including the 7.
Instead: (x + 5)/2 = 7 gives x + 5 = 14, so x = 9.

Show 8 more mistakes

Mistake: Forgetting to flip the sign: solving −2x > 10 as x > −5.
Dividing by a negative number reverses the order of the number line, so the inequality must reverse too.
Instead: −2x > 10 gives x < −5.
Mistake: Expanding 2(x + 3) as 2x + 3.
The 2 multiplies every term inside the bracket, not just the first.
Instead: 2(x + 3) = 2x + 6.
Mistake: Stopping at x and ignoring what the question asked, e.g. finding the son’s age when the father’s was asked.
The working earns marks, but the final answer must be the quantity requested, with units.
Instead: Re-read the last sentence before writing your final answer.
Mistake: Writing “x = 5, check: 3 × 5 − 4 = 11 ✓” without actually computing the check.
An unchecked substitution often hides a sign slip; examiners award the method mark but you lose the answer mark.
Instead: Do the arithmetic of the check visibly: 15 − 4 = 11 ✓.
Mistake: Treating x < 4 and x ≤ 4 as the same answer.
The first excludes 4; the second includes it. On a number line, one draws an open dot, the other a closed dot.
Instead: Match the endpoint treatment to the symbol in the question and in your answer.
Mistake: Solving 2x + 5 = 17 by adding 5: 2x = 22, x = 11.
The equation shows “+ 5”, so the inverse operation is to subtract 5 from both sides — adding moves the wrong way.
Instead: 2x = 12, so x = 6. Check: 2 × 6 + 5 = 17 ✓.
Mistake: Clearing a fraction in only one term: x/2 + x/3 = 5 becomes x + x/3 = 10.
Multiplying both sides by 6 must hit every term — the x/3 was left unscaled, so the equation changed.
Instead: Multiply every term: 3x + 2x = 30, so 5x = 30 and x = 6.
Mistake: Translating “at most 5” as x > 5.
“At most” allows the boundary value itself, so 5 is a valid answer; x > 5 throws it away.
Instead: “At most 5” means x ≤ 5; “at least 5” means x ≥ 5.

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

Solve one-step linear equations

Solve 3x = 21.

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

6Foundation
5Standard
4Challenging

15 questions worth 51 marks, each with a model answer, mark scheme and examiner tips.

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Exam 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 5 Practice Paper

30 marks · 40 min · 5 pages

10 multiple-choice questions and structured questions with answer lines. Chapter 5.

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Mark scheme

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Chapter 5 Practice Paper — mark scheme

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Model answers with a mark for each creditworthy point — for marking at home or in class.

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Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Term 2Coming soonWeighted Assessment 2 (WA2)
Term 4Coming soonEnd-of-Year Examination (EOY)

Questions

Frequently asked questions

What is Sec 1 Math Chapter 5 about?

Chapter 5 covers solving linear equations in one variable — including bracketed and fractional equations — forming equations from word problems, and solving simple linear inequalities with the sign-flip rule for negatives.

How do I solve an equation with a fraction like (2x − 1)/3 = 5?

Multiply both sides by the denominator: 2x − 1 = 15. Then solve the ordinary two-step equation: 2x = 16, so x = 8. Always check by substituting back into the original fraction.

When do I flip the inequality sign?

Only when you multiply or divide both sides by a negative number. Adding, subtracting, or scaling by a positive number keeps the sign as it is. Example: −2x > 10 becomes x < −5.

How do I turn a word problem into an equation?