What are inequalities and how do they differ from equations in Edexcel IGCSE Mathematics?

Inequalities are mathematical expressions that show the relationship between two values where one is not equal to the other, using symbols like , ≤, or ≥. Unlike equations, which state that two expressions are equal, inequalities indicate that one side is either greater than or less than the other. Understanding inequalities is crucial in the Edexcel IGCSE Mathematics curriculum as they help in solving real-world problems where exact values are not always possible.

How do you solve linear inequalities in one variable for the Edexcel IGCSE exam?

To solve linear inequalities in one variable, follow similar steps as solving linear equations: isolate the variable on one side of the inequality. However, remember that if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, if you have -2x > 6, dividing both sides by -2 gives x < -3. This rule is crucial for solving inequalities correctly in the Edexcel IGCSE Mathematics exam.

What is the importance of graphing inequalities on a number line in the Edexcel IGCSE syllabus?

Graphing inequalities on a number line is important as it visually represents the solution set of an inequality. This helps students understand the range of possible solutions. For instance, if x > 3, you would draw an open circle at 3 and shade to the right, indicating all numbers greater than 3. This skill is emphasized in the Edexcel IGCSE Mathematics curriculum to aid in comprehending and communicating solutions effectively.

Can you explain how to solve and graph compound inequalities for the Edexcel IGCSE Mathematics exam?

Compound inequalities involve two separate inequalities joined by 'and' or 'or'. For 'and' inequalities, find the intersection of the solutions, while for 'or' inequalities, find the union. For example, solving 1 < x + 2 ≤ 5 involves solving 1 < x + 2 and x + 2 ≤ 5 separately, then finding the overlap. Graphing these on a number line helps visualize the solution set, a key skill for the Edexcel IGCSE Mathematics exam.

What are some common mistakes to avoid when working with inequalities in Edexcel IGCSE Mathematics?

Common mistakes include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, misinterpreting the inequality symbols, and incorrectly graphing the solution on a number line. Additionally, students sometimes confuse the rules for solving equations with those for inequalities. Being mindful of these pitfalls is essential for success in the Edexcel IGCSE Mathematics curriculum.

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

Why it happens: Treating it like an equation. How to avoid it: ALWAYS flip when MULTIPLYING or DIVIDING both sides by a NEGATIVE. Highlight this step in working. Source: 4MA1/1H Jan 2024 — examiner report Q7

Why is "Wrong region for quadratic inequality" a common mistake in Inequalities?

Why it happens: Confusing 0. How to avoid it: Test a value in each region. Or sketch the parabola: above x-axis = positive, below = negative.

Why is "Including/excluding the boundary incorrectly" a common mistake in Inequalities?

Why it happens: EXCLUDE; , INCLUDE. Be explicit when listing integer solutions.

Equations, Formulae and IdentitiesEdexcel IGCSE Mathematics (4MA1)

Inequalities

Q: What is the importance of graphing inequalities on a number line in the Edexcel IGCSE syllabus?

Graphing inequalities on a number line is important as it visually represents the solution set of an inequality. This helps students understand the range of possible solutions. For instance, if x > 3, you would draw an open circle at 3 and shade to the right, indicating all numbers greater than 3. This skill is emphasized in the Edexcel IGCSE Mathematics curriculum to aid in comprehending and communicating solutions effectively.

Q: What are some common mistakes to avoid when working with inequalities in Edexcel IGCSE Mathematics?

Common mistakes include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, misinterpreting the inequality symbols, and incorrectly graphing the solution on a number line. Additionally, students sometimes confuse the rules for solving equations with those for inequalities. Being mindful of these pitfalls is essential for success in the Edexcel IGCSE Mathematics curriculum.

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

Why it happens: Treating it like an equation. How to avoid it: ALWAYS flip when MULTIPLYING or DIVIDING both sides by a NEGATIVE. Highlight this step in working. Source: 4MA1/1H Jan 2024 — examiner report Q7

Q: Why is "Wrong region for quadratic inequality" a common mistake in Inequalities?

Why it happens: Confusing 0. How to avoid it: Test a value in each region. Or sketch the parabola: above x-axis = positive, below = negative.

Q: Why is "Including/excluding the boundary incorrectly" a common mistake in Inequalities?

Why it happens: EXCLUDE; , INCLUDE. Be explicit when listing integer solutions.

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Inequalities Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Linear and quadratic inequalities. Number-line representation. Critical: flip when multiplying/dividing by a negative.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

2.8 — Solve linear inequalities.
2.8 — Solve quadratic inequalities.
2.8 — Represent solutions on number lines.
2.8 — Find integer solutions in a range.

Linear inequalities

Solve like equations. FLIP when × or ÷ by negative.

Standard methods (same as for equations) EXCEPT:

When MULTIPLYING or DIVIDING both sides by a NEGATIVE, FLIP THE INEQUALITY.
Adding/subtracting: NO flip.
Multiplying/dividing by POSITIVE: NO flip.

Example. $3x - 5 < 13$ .

Add $5$ : $3x < 18$ .
Divide by $3$ : $x < 6$ . (Positive; no flip.)

Example with flip. $-2x + 7 \le 1$ .

Subtract $7$ : $-2x \le -6$ .
Divide by $-2$ AND FLIP: $x \ge 3$ .

Verify. Test $x = 3$ : $-2(3) + 7 = 1 \le 1$ ✓. Test $x = 4$ : $-1 \le 1$ ✓. Test $x = 2$ : $3 \le 1$ ✗. So $x \ge 3$ is correct.

Double inequalities.

$-7 < 2x - 3 \le 5$ . Apply each operation to ALL parts.

Add $3$ : $-4 < 2x \le 8$ .
Divide by $2$ : $-2 < x \le 4$ .

Why does multiplying by a negative flip the sign?

$2 < 3$ is true. Multiply both by $-1$ : $-2$ vs $-3$ . Now $-2 > -3$ . The inequality flipped.

Edexcel tip. When you flip, write 'flipping inequality due to negative divisor' as a side note. This earns the M1.

Linear: same as equations except for negatives.
FLIP when × or ÷ by negative.
Double inequality: apply to all three parts.
Test values to verify.

Quadratic inequalities

Factor, find roots, test regions. 'Between' = product negative.

Method:

Rearrange to $ax^{2} + bx + c < 0$ (or $> 0$ , $\le$ , $\ge$ ).
Factor (or use quadratic formula).
Find the ROOTS (where the quadratic = 0).
Test a value in each region of the number line.
State the solution as an interval.

Example. $x^{2} - 5x - 14 < 0$ .

Already in standard form.
Factor: $(x - 7)(x + 2) < 0$ .
Roots: $x = -2, x = 7$ . These divide the number line into three regions.

Region	Test value	$(x-7)(x+2)$	Sign
$x < -2$	$-3$	$(-10)(-1) = 10$	+
$-2 < x < 7$	$0$	$(-7)(2) = -14$	−
$x > 7$	$8$	$(1)(10) = 10$	+

We want NEGATIVE, so the answer is the middle region: $-2 < x < 7$ .

Sketch method (for visual learners).

Plot the parabola $y = x^{2} - 5x - 14$ :

Roots at $x = -2, 7$ .
Up-opening parabola (positive $x^{2}$ coefficient).
BELOW the $x$ -axis BETWEEN the roots.
ABOVE the $x$ -axis OUTSIDE the roots.

Match to the inequality direction.

Common patterns:

Form	Solution shape
$(x - a)(x - b) < 0$ , $a < b$	$a < x < b$ (between)
$(x - a)(x - b) > 0$ , $a < b$	$x < a$ OR $x > b$ (outside)

Worked qualitative. Why does 'between' correspond to negative product?

Two factors with opposite signs multiply to negative.
One factor positive (right of one root) AND one negative (left of other) → between the roots.
Both positive or both negative → outside.

Edexcel tip. Always state the FACTORS and the ROOTS explicitly. Then state the solution interval. Mark schemes credit each step.

Factor and find roots.
Test regions on number line.
Sketch parabola if helps.
Negative product: between roots.
Positive product: outside roots.

Number-line representation

Open circle for strict ( $<, >$ ), closed for inclusive ( $\le, \ge$ ).

Conventions:

$<$ or $>$ : OPEN CIRCLE (boundary excluded).
$\le$ or $\ge$ : CLOSED CIRCLE (boundary included).
Arrow shows the direction.

Example. $x < 6$ :

←——————————o
            6

Open circle at $6$ , arrow extending left.

Example. $-2 < x \le 4$ :

       o——————•
       -2     4

Open at $-2$ (excluded), closed at $4$ (included), shaded between.

Set notation.

Sometimes Edexcel uses set-builder: $\{x : -2 < x \le 4\}$ .

Or interval notation: $(-2, 4]$ . Round bracket = excluded; square = included.

Worked qualitative. What's the difference between $\{x : x \le 5\}$ and $\{x : x < 5\}$ for INTEGERS?

$x \le 5$ : $..., 3, 4, 5$ .
$x < 5$ : $..., 3, 4$ .
$5$ included for $\le$ , excluded for $<$ .

Edexcel tip. Match the bracket type to the inequality. Strict → open. Inclusive → closed.

Open circle: strict.
Closed circle: inclusive.
Number line shows direction.
Set notation: $\{x : ...\}$ .

Listing integer solutions

Solve, then list integers within range.

Method:

Solve the inequality (all the rules apply).
List all INTEGERS within the resulting range.

Example. $-3 < 2x - 1 \le 5$ .

Add $1$ : $-2 < 2x \le 6$ . Divide by $2$ : $-1 < x \le 3$ .
Integers: $0, 1, 2, 3$ .

(Note: $-1$ excluded by strict inequality; $3$ included.)

Quadratic example. Find integer values of $x$ for which $x^{2} - 4x \le 0$ .

Factor: $x(x - 4) \le 0$ .
Roots: $0, 4$ . Between them = product non-positive.
So $0 \le x \le 4$ .
Integers: $0, 1, 2, 3, 4$ .

Worked qualitative. What if the range includes a fractional boundary?

$-1.5 < x \le 3.7$ : integers are $-1, 0, 1, 2, 3$ .
Round upward for the lower bound (next integer at or above $-1.5$ is $-1$ ).
Round downward for the upper bound (last integer at or below $3.7$ is $3$ ).

Edexcel tip. Write all integers explicitly. Don't shortcut to $\{0, 1, 2, 3\}$ — list each separately for clarity.

Solve first.
List integers in range.
Strict bounds: exclude.
Inclusive bounds: include.

How it’s examined

Inequalities appear Paper 1H + 2H (3-5 marks). Linear (2-3 marks), quadratic for A* candidates (4-5 marks). Examiner reports flag (1) not flipping for negative, (2) wrong region for quadratic, (3) confusion about strict vs inclusive boundaries.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H Jan 2024, 4MA1/1H May/Jun 2024. Last reviewed 2026-05-07.

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Step-by-step worked examples — Inequalities

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

1Solve a linear inequality
Foundation• inequality, linear
Question
Solve $3x - 5 < 13$ .
Step-by-step solution
1. Step 1
  Add $5$ : $3x < 18$ .
2. Step 2
  Divide by $3$ : $x < 6$ .
Answer
$x < 6$
2Flip when multiplying/dividing by negative
Foundation• Adapted from 4MA1/1H Jan 2024 Q7• inequality, flip
Question
Solve $-2x + 7 \le 1$ .
Step-by-step solution
1. Step 1
  Subtract $7$ : $-2x \le -6$ .
2. Step 2
  Divide by $-2$ . FLIP THE INEQUALITY when dividing by negative:
  $x \ge 3$
Answer
$x \ge 3$
Examiner tip
ALWAYS flip the inequality when multiplying or dividing both sides by a NEGATIVE number. Mark scheme awards M1 for correct flipping.
3Double inequality
Foundation• double inequality
Question
Solve $-7 < 2x - 3 \le 5$ .
Step-by-step solution
1. Step 1
  Add $3$ to all three parts: $-4 < 2x \le 8$ .
2. Step 2
  Divide by $2$ : $-2 < x \le 4$ .
Answer
$-2 < x \le 4$
Examiner tip
Apply each operation to ALL THREE parts simultaneously.
4Quadratic inequality
Higher• Adapted from 4MA1/1H May/Jun 2024 Q19• quadratic inequality
Question
Solve $x^{2} - 5x - 14 < 0$ .
Step-by-step solution
1. Step 1
  Factor: $(x - 7)(x + 2) < 0$ .
2. Step 2
  Roots: $x = 7$ and $x = -2$ . These divide the number line into three intervals.
3. Step 3
  Test a value in each interval:
4. Step 4
  $x = -3$ : $(-10)(-1) = +10 > 0$ . $x = 0$ : $(-7)(2) = -14 < 0$ ✓. $x = 8$ : $(1)(10) = +10 > 0$ .
5. Step 5
  So $(x - 7)(x + 2) < 0$ when $-2 < x < 7$ .
Answer
$-2 < x < 7$
Examiner tip
For quadratic inequalities, factor THEN sketch a parabola or test values. The 'between' answer occurs when product is negative.
5Integer solutions
Higher• integer solutions
Question
List the integer solutions of $-3 < 2x - 1 \le 5$ .
Step-by-step solution
1. Step 1
  Add $1$ : $-2 < 2x \le 6$ . Divide by $2$ : $-1 < x \le 3$ .
2. Step 2
  Integer values: $x = 0, 1, 2, 3$ .
Answer
$x = 0, 1, 2, 3$
Examiner tip
Integer solutions: list ALL integers in the range. $-1 < x$ : doesn't include $-1$ . $x \le 3$ : includes $3$ .

Key Definitions and Keywords — Inequalities

Definitions to memorise and the exact keywords mark schemes credit for inequalities answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Inequality
Examiner keyword
A statement comparing two quantities using $<$ , $>$ , $\le$ , or $\ge$ .
Example
$x < 5$ , $-2 \le y \le 3$ .
Strict ( $<$ , $>$ ) vs Inclusive ( $\le$ , $\ge$ )
Strict inequalities EXCLUDE the boundary; inclusive INCLUDE it.
Example
$x < 5$ excludes $5$ . $x \le 5$ includes $5$ .

Common Mistakes and Misconceptions — Inequalities

The traps other students keep falling into on inequalities questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Not flipping the inequality when dividing by a negative
4MA1/1H Jan 2024 — examiner report Q7
Why it happens
Treating it like an equation.
How to avoid it
ALWAYS flip when MULTIPLYING or DIVIDING both sides by a NEGATIVE. Highlight this step in working.
✕Wrong region for quadratic inequality
Why it happens
Confusing $< 0$ with $> 0$ .
How to avoid it
Test a value in each region. Or sketch the parabola: above $x$ -axis = positive, below = negative.
✕Including/excluding the boundary incorrectly
Why it happens
$<$ vs $\le$ confusion.
How to avoid it
$<$ , $>$ EXCLUDE; $\le$ , $\ge$ INCLUDE. Be explicit when listing integer solutions.

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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Video lesson

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

Inequalities — frequently asked questions

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

Inequalities

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Solve a linear inequality

2Flip when multiplying/dividing by negative

3Double inequality

4Quadratic inequality

5Integer solutions

Inequality

Strict (<<<, >>>) vs Inclusive (≤\le≤, ≥\ge≥)

✕Not flipping the inequality when dividing by a negative

✕Wrong region for quadratic inequality

✕Including/excluding the boundary incorrectly

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Inequalities — frequently asked questions

Strict ( $<$ , $>$ ) vs Inclusive ( $\le$ , $\ge$ )