What is a linear equation and how is it different from a linear inequality?

A linear equation is a mathematical statement that shows the equality of two expressions, typically in the form ax + b = c, where a, b, and c are constants. A linear inequality, on the other hand, expresses a relationship where one side is not equal to the other, using symbols like , ≤, or ≥. For example, ax + b < c represents a linear inequality. Both are fundamental concepts in the Cambridge IGCSE Mathematics curriculum under Algebra.

How do you solve a linear equation in one variable?

To solve a linear equation in one variable, such as ax + b = c, you need to isolate the variable x. Start by subtracting b from both sides to get ax = c - b. Then, divide both sides by a to solve for x, resulting in x = (c - b)/a. This process is a key skill in the Cambridge IGCSE Mathematics syllabus.

What are the steps to graph a linear inequality on a coordinate plane?

To graph a linear inequality, first graph the corresponding linear equation as if the inequality were an equality (e.g., y = mx + c). Use a solid line for ≤ or ≥, and a dashed line for . Next, choose a test point not on the line to determine which side of the line satisfies the inequality. Shade the region that satisfies the inequality. Understanding this process is crucial for Cambridge IGCSE Mathematics students.

Can you explain how to solve a system of linear equations?

A system of linear equations involves finding the values of variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing. In substitution, solve one equation for a variable and substitute it into the other equation. In elimination, add or subtract equations to eliminate a variable. Graphing involves plotting both equations and finding their intersection point. Mastery of these methods is essential for Cambridge IGCSE Mathematics.

What are some real-life applications of linear equations and inequalities?

Linear equations and inequalities are used in various real-life scenarios, such as budgeting, where they help in determining costs and profits, or in physics, to calculate speed and distance. They are also used in business for optimizing resources and in computer science for algorithms. Understanding these applications is part of the Cambridge IGCSE Mathematics curriculum, highlighting the practical importance of algebra.

Why is "Not flipping the inequality when dividing by a negative" a common mistake in Linear Equations and Inequalities?

Why it happens: Students treat inequalities like equations and forget the sign-flip rule. How to avoid it: Memorise: divide/multiply by negative → flip the inequality. Always. Source: 0580 examiner reports — every series

Why is "Distributing only the first term across a bracket" a common mistake in Linear Equations and Inequalities?

Why it happens: Speed-running expansions: 5(2x - 3) becomes 10x - 3 instead of 10x - 15. How to avoid it: Multiply EVERY term inside the bracket by the factor outside.

Why is "Multiplying only some terms by the LCM when clearing fractions" a common mistake in Linear Equations and Inequalities?

Why it happens: Students multiply the fraction terms but forget the constant on the right-hand side. How to avoid it: Apply × 12 (or whatever the LCM is) to **every term** in the equation.

Why is "Wrong open/closed circle on a number line" a common mistake in Linear Equations and Inequalities?

Why it happens: Strict ( ) and non-strict (≤/≥) inequalities use different markers and students confuse them. How to avoid it: → open circle (excluded). ≤ or ≥ → closed (filled) circle (included).

AlgebraCambridge IGCSE Mathematics (0580)

Linear Equations and Inequalities

Q: How do you solve a linear equation in one variable?

To solve a linear equation in one variable, such as ax + b = c, you need to isolate the variable x. Start by subtracting b from both sides to get ax = c - b. Then, divide both sides by a to solve for x, resulting in x = (c - b)/a. This process is a key skill in the Cambridge IGCSE Mathematics syllabus.

Q: What are the steps to graph a linear inequality on a coordinate plane?

To graph a linear inequality, first graph the corresponding linear equation as if the inequality were an equality (e.g., y = mx + c). Use a solid line for ≤ or ≥, and a dashed line for . Next, choose a test point not on the line to determine which side of the line satisfies the inequality. Shade the region that satisfies the inequality. Understanding this process is crucial for Cambridge IGCSE Mathematics students.

Q: Can you explain how to solve a system of linear equations?

A system of linear equations involves finding the values of variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing. In substitution, solve one equation for a variable and substitute it into the other equation. In elimination, add or subtract equations to eliminate a variable. Graphing involves plotting both equations and finding their intersection point. Mastery of these methods is essential for Cambridge IGCSE Mathematics.

Q: What are some real-life applications of linear equations and inequalities?

Linear equations and inequalities are used in various real-life scenarios, such as budgeting, where they help in determining costs and profits, or in physics, to calculate speed and distance. They are also used in business for optimizing resources and in computer science for algorithms. Understanding these applications is part of the Cambridge IGCSE Mathematics curriculum, highlighting the practical importance of algebra.

Q: Why is "Not flipping the inequality when dividing by a negative" a common mistake in Linear Equations and Inequalities?

Why it happens: Students treat inequalities like equations and forget the sign-flip rule. How to avoid it: Memorise: divide/multiply by negative → flip the inequality. Always. Source: 0580 examiner reports — every series

Q: Why is "Distributing only the first term across a bracket" a common mistake in Linear Equations and Inequalities?

Why it happens: Speed-running expansions: 5(2x - 3) becomes 10x - 3 instead of 10x - 15. How to avoid it: Multiply EVERY term inside the bracket by the factor outside.

Q: Why is "Multiplying only some terms by the LCM when clearing fractions" a common mistake in Linear Equations and Inequalities?

Why it happens: Students multiply the fraction terms but forget the constant on the right-hand side. How to avoid it: Apply × 12 (or whatever the LCM is) to **every term** in the equation.

Q: Why is "Wrong open/closed circle on a number line" a common mistake in Linear Equations and Inequalities?

Why it happens: Strict ( ) and non-strict (≤/≥) inequalities use different markers and students confuse them. How to avoid it: → open circle (excluded). ≤ or ≥ → closed (filled) circle (included).

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Linear Equations and Inequalities — Cambridge IGCSE 0580 Maths Extended (2026)

Solve equations of the form $ax + b = c$ and inequalities like $2x - 3 \le 7$ . The same balancing rules in both — except multiplying or dividing an inequality by a negative FLIPS the sign.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E2.7 — Construct and solve linear equations.
E2.8 — Solve linear inequalities and represent solutions on a number line.

Solving linear equations

Reverse BIDMAS to isolate $x$ . Whatever you do to one side, do to the other.

Method.

Expand any brackets.
Move all $x$ -terms to one side, all numbers to the other.
Combine like terms.
Divide by the coefficient of $x$ .

Worked. Solve $3x + 7 = 22$ .

Subtract $7$ : $3x = 15$ .
Divide by $3$ : $x = 5$ .

Worked. Solve $5x - 3 = 2x + 9$ .

Subtract $2x$ : $3x - 3 = 9$ .
Add $3$ : $3x = 12$ .
Divide by $3$ : $x = 4$ .

Worked (with brackets). Solve $4(x + 2) = 3(2x - 1)$ .

Expand: $4x + 8 = 6x - 3$ .
Subtract $4x$ : $8 = 2x - 3$ .
Add $3$ : $11 = 2x$ .
Divide: $x = 5.5$ .

Worked (with fractions). Solve $\dfrac{x + 3}{2} = \dfrac{2x - 1}{3}$ .

Cross-multiply: $3(x + 3) = 2(2x - 1)$ .
Expand: $3x + 9 = 4x - 2$ .
Subtract $3x$ : $9 = x - 2$ .
Add $2$ : $x = 11$ .

Always check by substituting back.

Expand brackets first.
Variables to one side, numbers to the other.
Same operation on BOTH sides.
Cross-multiply to clear two-sided fractions.
Always check by substituting back.

Solving linear inequalities

Same balancing as equations — until you multiply or divide by a negative. Then flip the sign.

Linear inequalities are solved with the same procedure as equations, with ONE crucial twist:

The flip rule. Multiplying or dividing both sides by a NEGATIVE number REVERSES the inequality. $\text{If } a < b \text{ and } c < 0, \text{ then } ac > bc.$

Adding, subtracting, and multiplying/dividing by a positive NEVER flip the sign.

Worked. Solve $2x + 5 < 13$ .

Subtract $5$ : $2x < 8$ .
Divide by $2$ (positive — no flip): $x < 4$ .

Worked. Solve $5 - 3x \ge 11$ .

Subtract $5$ : $-3x \ge 6$ .
Divide by $-3$ — FLIP: $x \le -2$ .

Worked. Solve $4x - 7 \ge 2x + 1$ .

Subtract $2x$ : $2x - 7 \ge 1$ .
Add $7$ : $2x \ge 8$ .
Divide by $2$ : $x \ge 4$ .

Tip. To avoid the flip, move the variable to whichever side keeps it positive. For $5 - 3x \ge 11$ , move $-3x$ across to give $5 \ge 11 + 3x$ , then $-6 \ge 3x$ , then $x \le -2$ . Same answer, no flip needed.

Same balancing as equations, plus the flip rule.
Multiply or divide by NEGATIVE → flip the inequality.
Adding/subtracting never flips.
To avoid the flip, keep the variable on the positive side.

Number-line representation

Use a circle (open or closed) and a directed line to show the solution.

Inequalities can be drawn on a number line with these conventions:

Symbol	Endpoint	Direction
$x < 4$	OPEN circle at $4$	Arrow LEFT
$x \le 4$	CLOSED circle at $4$	Arrow LEFT
$x > -1$	OPEN circle at $-1$	Arrow RIGHT
$x \ge -1$	CLOSED circle at $-1$	Arrow RIGHT

For a double inequality $-2 \le x < 5$ :

CLOSED circle at $-2$ , OPEN circle at $5$ .
Solid line BETWEEN them.

Integer solutions. Some questions ask for integer solutions only. From $-2 \le x < 5$ , the integers are $\{-2, -1, 0, 1, 2, 3, 4\}$ — note that $5$ is excluded (open circle), $-2$ is included (closed).

$<, >$ → OPEN circle.
$\le, \ge$ → CLOSED circle.
Direction = arrow.
Double inequality: line between two endpoints, each circle styled accordingly.

How it’s examined

Linear equations are guaranteed on every paper. Paper 2 has them as 1-2 mark single-step questions; Paper 4 embeds them inside word problems (3-5 marks). Linear inequalities appear most years as 2-3 mark items, often with a number-line representation. Examiner reports flag forgetting to flip the sign and miscounting integer solutions on inclusive/exclusive endpoints.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E2.7-2.8); 0580/22 May/Jun 2024 — Q9 (linear equation); 0580/42 Oct/Nov 2024 — Q7 (inequality with flip); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Linear Equations and Inequalities

Step-by-step solutions to past-paper-style questions on linear equations and inequalities, written exactly the way a tutor would explain them at the board.

1Solve a linear equation
Core• linear equation
Question
Solve $5(2x - 3) = 4x + 9$ .
Step-by-step solution
1. Step 1
  Expand the bracket.
  $10x - 15 = 4x + 9$
2. Step 2
  Move $x$ terms to the left.
  $10x - 4x = 9 + 15$
3. Step 3
  Simplify.
  $6x = 24$
4. Step 4
  Divide by $6$ .
  $x = 4$
Answer
$x = 4$
Examiner tip
Always check by substituting back: $5(2 \cdot 4 - 3) = 5(5) = 25$ and $4(4) + 9 = 25$ . ✓
2Solve an equation with fractions
Extended• Adapted from 0580/22 Oct/Nov 2023 Q9• fractions
Question
Solve $\dfrac{x + 1}{3} - \dfrac{x - 2}{4} = 2$ .
Step-by-step solution
1. Step 1
  Multiply every term by $12$ (LCM of $3$ and $4$ ).
  $4(x + 1) - 3(x - 2) = 24$
2. Step 2
  Expand.
  $4x + 4 - 3x + 6 = 24$
3. Step 3
  Collect like terms.
  $x + 10 = 24$
4. Step 4
  Solve.
  $x = 14$
Answer
$x = 14$
Examiner tip
Multiplying through by the LCM clears all fractions in one step. Watch the sign on $-3(x-2)$ : it gives $-3x + 6$ , not $-3x - 6$ .
3Solve a linear inequality
Core• inequality
Question
Solve $3x - 4 \leq 11$ and represent the solution on a number line.
Step-by-step solution
1. Step 1
  Add $4$ .
  $3x \leq 15$
2. Step 2
  Divide by $3$ .
  $x \leq 5$
3. Step 3
  On a number line: solid (filled) circle at $5$ , arrow pointing left (because $x$ is less-than-or-equal).
Answer
$x \leq 5$
4Inequality with a negative coefficient
Core• inequality, flip
Question
Solve $7 - 2x > 1$ .
Step-by-step solution
1. Step 1
  Subtract $7$ .
  $-2x > -6$
2. Step 2
  Divide by $-2$ — flip the inequality.
  $x < 3$
Answer
$x < 3$
Examiner tip
Multiplying or dividing by a negative number REVERSES the inequality. Forgetting this is the single most common error in inequality questions across recent series.
5Solve a double inequality
Extended• double inequality
Question
Solve $-3 \leq 2x - 1 < 5$ .
Step-by-step solution
1. Step 1
  Apply each operation to all three parts simultaneously. Add $1$ .
  $-2 \leq 2x < 6$
2. Step 2
  Divide by $2$ .
  $-1 \leq x < 3$
Answer
$-1 \leq x < 3$

Key Formulae — Linear Equations and Inequalities

The formulae you need to memorise for linear equations and inequalities on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Linear-equation strategy
$ax + b = c \implies x = \dfrac{c - b}{a}$
$a, b, c$
constants with $a \neq 0$
When to use
Standard one-step solve once you have the linear equation in this form.
Inequality-flip rule
$\text{If } a < b \text{ and } k < 0,\ \text{then } ka > kb$
When to use
Whenever you multiply or divide both sides of an inequality by a negative number — flip the sign.

Key Definitions and Keywords — Linear Equations and Inequalities

Definitions to memorise and the exact keywords mark schemes credit for linear equations and inequalities answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Equation
Examiner keyword
A statement of equality between two algebraic expressions, true only for specific values of the variable.
Inequality
Examiner keyword
A statement using $<$ , $>$ , $\leq$ or $\geq$ instead of $=$ . The solution is a range of values.
Solution set
The set of all values of the variable that satisfy the equation or inequality.
Strict / non-strict inequality
Examiner keyword
Strict ( $<$ , $>$ ) excludes the endpoint; non-strict ( $\leq$ , $\geq$ ) includes it.
Number line representation
A line with arrow showing the direction of the solution, with an open circle for strict inequalities and a closed (filled) circle for non-strict.

Common Mistakes and Misconceptions — Linear Equations and Inequalities

The traps other students keep falling into on linear equations and inequalities questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Not flipping the inequality when dividing by a negative
0580 examiner reports — every series
Why it happens
Students treat inequalities like equations and forget the sign-flip rule.
How to avoid it
Memorise: divide/multiply by negative → flip the inequality. Always.
✕Distributing only the first term across a bracket
Why it happens
Speed-running expansions: $5(2x - 3)$ becomes $10x - 3$ instead of $10x - 15$ .
How to avoid it
Multiply EVERY term inside the bracket by the factor outside.
✕Multiplying only some terms by the LCM when clearing fractions
Why it happens
Students multiply the fraction terms but forget the constant on the right-hand side.
How to avoid it
Apply $\times 12$ (or whatever the LCM is) to every term in the equation.
✕Wrong open/closed circle on a number line
Why it happens
Strict ( $<$ / $>$ ) and non-strict ( $\leq$ / $\geq$ ) inequalities use different markers and students confuse them.
How to avoid it
$<$ or $>$ → open circle (excluded). $\leq$ or $\geq$ → closed (filled) circle (included).

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Short walkthrough of the concepts students most often get stuck on.

Linear Equations and Inequalities — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Linear Equations and Inequalities

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Solve a linear equation

2Solve an equation with fractions

3Solve a linear inequality

4Inequality with a negative coefficient

5Solve a double inequality

Linear-equation strategy

Inequality-flip rule

Equation

Inequality

Solution set

Strict / non-strict inequality

Number line representation

✕Not flipping the inequality when dividing by a negative

✕Distributing only the first term across a bracket

✕Multiplying only some terms by the LCM when clearing fractions

✕Wrong open/closed circle on a number line

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Linear Equations and Inequalities — frequently asked questions