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).

Cambridge 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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Short Notes - Linear Equations and Inequalities

Detailed Study Notes

Detailed notes on Algebra for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

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.

At a glance

An equation has $=$ ; an inequality has $<$ , $\le$ , $>$ or $\ge$ .
Same operation on both sides preserves the truth.
Reverse BIDMAS to isolate the variable.
Inequality flip: multiplying or dividing by a NEGATIVE flips $<$ to $>$ (and vice versa).
Adding/subtracting NEVER flips an inequality.
Inequality solutions can be a single value, a single range, or two pieces.
Number-line representation: closed dot for $\le, \ge$ ; open dot for $<, >$ .

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 $3$ : $x = 5$ .

The two sides stay equal as long as every operation is applied to both.

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.

Multiplying by a negative reflects both numbers about zero, so their order — and the inequality — reverses.

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: $x \le -2$ .

Dividing both sides by −3 reverses ≥ into ≤ — the one twist inequalities add.

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.

Closed circle for ≤ or ≥ (endpoint included); open circle for < or > (endpoint excluded).

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.

Quick recap

Equations: balance both sides until $x$ is isolated.
Cross-multiply to clear two-sided fractions.
Inequality method = equation method PLUS the flip rule.
Multiply or divide by negative → reverse the inequality.
Number line: open circle for strict, closed for non-strict.
Always state integer-only solutions when asked.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Linear equation — equation in which the variable appears only to the first power.
Linear inequality — inequality of the same form: $ax + b < c$ etc.
Inequality flip rule — multiplying or dividing both sides by a negative reverses the direction.

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.

Take this whole topic with you

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.

Question type:

Question patterns to master — Linear Equations and Inequalities

Almost every linear equations and inequalities exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Direct calculation

Recognise it by

A single instruction — solve — on a linear equation or inequality, sometimes with a number-line or integer-list request attached.

How to approach it

Expand brackets, clear fractions by multiplying every term by the LCM, collect the variable on one side, then divide. For inequalities, flip the sign whenever you multiply or divide by a negative.

Common trap

Forgetting to reverse the inequality when dividing by a negative — examiner reports flag this every series. For number lines, use an open circle for $<,>$ and a filled circle for $\le,\ge$ .

Word problem

Recognise it by

A real-world context — fares, budgets, at most, at least — with no equation or inequality written down. You must form it yourself.

How to approach it

Define the unknown with a letter, translate the context into an equation or inequality ( $\le$ for at most, $\ge$ for at least), solve it, then interpret the answer in context.

Common trap

Rounding the final answer the wrong way. With at most, round down to a whole number; rounding up breaks the constraint. Examiner reports flag this repeatedly.

1Solve a linear equation
CoreDirect calculation• 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
ExtendedDirect calculation• 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
CoreDirect calculation• 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
CoreDirect calculation• 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
ExtendedDirect calculation• 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
$-1 \leq x < 3$
6Brackets on both sides with variable on both sides
ExtendedDirect calculation• Adapted from 0580/22 Feb/Mar 2024 Q11• linear equation, brackets
Question
Solve $3(2x - 1) = 2(x + 5) - 4$ .
Step-by-step solution
1. Step 1
  Expand both sides.
  $6x - 3 = 2x + 10 - 4$
2. Step 2
  Simplify the right side.
  $6x - 3 = 2x + 6$
3. Step 3
  Move $x$ terms to the left, constants to the right.
  $6x - 2x = 6 + 3$
4. Step 4
  Simplify and solve.
  $4x = 9 \implies x = \dfrac{9}{4}$
Answer
$x = \dfrac{9}{4}$
Examiner tip
Examiners reward candidates who give exact fractions over rounded decimals. $x = 2.25$ scores the accuracy mark, but if the question says "exact value", only $\dfrac{9}{4}$ is accepted.
7Equation with a variable in two different fractions
ExtendedDirect calculation• Adapted from 0580/42 May/Jun 2024 Q10• fractions, variable both sides
Question
Solve $\dfrac{2x - 1}{3} = \dfrac{x + 4}{5}$ .
Step-by-step solution
1. Step 1
  Cross-multiply (multiply each side by $15$ ).
  $5(2x - 1) = 3(x + 4)$
2. Step 2
  Expand both sides.
  $10x - 5 = 3x + 12$
3. Step 3
  Collect.
  $10x - 3x = 12 + 5 \implies 7x = 17$
4. Step 4
  Solve.
  $x = \dfrac{17}{7}$
Answer
$x = \dfrac{17}{7}$
Examiner tip
The 2024 mark scheme awards method marks for cross-multiplying correctly. The dominant slip is multiplying numerators with denominators on the same side (e.g. $(2x-1) \times 3$ ) instead of across the equals sign.
8Compound inequality with mixed strict signs
ExtendedDirect calculation• Adapted from 0580/22 Oct/Nov 2024 Q11• compound inequality
Question
Solve $-4 < 3 - x \leq 7$ and list the integer solutions.
Step-by-step solution
1. Step 1
  Subtract $3$ from each part.
  $-7 < -x \leq 4$
2. Step 2
  Multiply each part by $-1$ — this REVERSES both inequality signs.
  $7 > x \geq -4$
3. Step 3
  Rewrite with the smaller number on the left.
  $-4 \leq x < 7$
4. Step 4
  Integer values in this range: $-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6$ .
Answer
$-4 \leq x < 7$ ; integers: $-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6$
Examiner tip
The examiner report flags candidates often forget that multiplying a compound inequality by $-1$ flips both signs. Verify the smaller bound is on the left at the end.
9Inequality involving a fraction with a negative coefficient
ExtendedDirect calculation• Adapted from 0580/42 Oct/Nov 2023 Q10• inequality, fractions, flip
Question
Solve $\dfrac{5 - 2x}{3} \geq 4$ .
Step-by-step solution
1. Step 1
  Multiply both sides by $3$ ( $3 > 0$ , so the sign does not flip).
  $5 - 2x \geq 12$
2. Step 2
  Subtract $5$ from both sides.
  $-2x \geq 7$
3. Step 3
  Divide both sides by $-2$ — REVERSE the inequality.
  $x \leq -\dfrac{7}{2}$
Answer
$x \leq -\dfrac{7}{2}$
Examiner tip
The examiner report flags candidates often flip the sign when multiplying by $3$ (no flip needed) but forget to flip when dividing by $-2$ . Track each operation's sign separately.
10Word problem leading to a linear inequality
ChallengeWord problem• Adapted from 0580/42 May/Jun 2023 Q17• word problem, inequality
Question
A taxi charges a $\$ 3 $fixed fee plus$ $1.20 $per kilometre. Maya has at most$ $25$ to spend. Find the maximum whole number of kilometres she can travel.
Step-by-step solution
1. Step 1
  Let $k$ be the number of kilometres. The total cost is $3 + 1.20k$ .
2. Step 2
  Set up an inequality: cost is at most $\$ 25$.
  $3 + 1.20k \leq 25$
3. Step 3
  Subtract $3$ .
  $1.20k \leq 22$
4. Step 4
  Divide by $1.20$ .
  $k \leq 18.333\dots$
5. Step 5
  Since $k$ must be a whole number and at most $18.33\dots$ , the maximum is $k = 18$ .
Answer
$18$ km
Examiner tip
The examiner report flags candidates often round up to $19$ . With "at most", the answer must be less than or equal to the bound — so round DOWN to the nearest integer.

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).

Linear Equations and Inequalities — frequently asked questions

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

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.

Multiplying by a negative reflects both numbers about zero, so their order — and the inequality — reverses.

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: $x \le -2$ .

Dividing both sides by −3 reverses ≥ into ≤ — the one twist inequalities add.

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

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

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

Linear Equations and Inequalities

Short Notes - Linear Equations and Inequalities

Detailed Study Notes

At a glance

What you’ll learn

Solving linear equations

Solving linear inequalities

Number-line representation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Linear Equations and Inequalities

1Solve a linear equation

2Solve an equation with fractions

3Solve a linear inequality

4Inequality with a negative coefficient

5Solve a double inequality

6Brackets on both sides with variable on both sides

7Equation with a variable in two different fractions

8Compound inequality with mixed strict signs

9Inequality involving a fraction with a negative coefficient

10Word problem leading to a linear 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

Linear Equations and Inequalities — frequently asked questions

Linear Equations and Inequalities

Short Notes - Linear Equations and Inequalities

Detailed Study Notes

At a glance

What you’ll learn

Solving linear equations

Solving linear inequalities

Number-line representation

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Linear Equations and Inequalities

1Solve a linear equation

2Solve an equation with fractions

3Solve a linear inequality

4Inequality with a negative coefficient

5Solve a double inequality

6Brackets on both sides with variable on both sides

7Equation with a variable in two different fractions

8Compound inequality with mixed strict signs

9Inequality involving a fraction with a negative coefficient

10Word problem leading to a linear 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

Linear Equations and Inequalities — frequently asked questions