Back

Inequalities and Number Line Tricks for O Level & IGCSE Mathematics

Views 1

Understanding Inequality Symbols With Absolute Clarity

  • Inequality vocabulary

    • < means “less than”

    • ≤ means “less than or equal to”

    • means “greater than”

    • ≥ means “greater than or equal to”

  • Always check the direction of arrows

  • Inequality set notation lines up smallest value on left

  • Inequalities describe ranges, not single values

  • Open vs closed conditions define whether endpoint included

Solving Linear Inequalities Step by Step

  • Treat inequality like equation, except for sign flips

  • Add or subtract same number on both sides

  • Multiply or divide both sides by positive value → sign unchanged

  • Multiply or divide both sides by negative value → SIGN REVERSES

  • Always isolate the unknown

  • Re-order inequality final answer with smallest number on left

  • Check for operations on both sides

  • Avoid mistakes with negative multipliers

Examples:

  • 3x + 5 < 20

    • 3x < 15

    • x < 5

  • −2x ≥ 14

    • x ≤ −7 (sign flips)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Compound Inequalities (Double Inequalities)

  • Form like: a < x < b

  • Solve by performing same operation on all three sides

  • Keep inequality orientation consistent

  • When multiplying or dividing by negative → flip BOTH signs

Example:

  • −2 ≤ 3x + 1 < 10

    • Subtract 1: −3 ≤ 3x < 9

    • Divide by 3: −1 ≤ x < 3

  • Represent result as number line range

Number Line Basics to Avoid Presentation Errors

  • Number line requires clear marking

  • Open circle for: < or >

  • Closed circle for: ≤ or ≥

  • Shade direction

    • Shade rightwards for x > a

    • Shade leftwards for x < a

  • When range given, shade between values

  • Use neat, equally spaced tick marks

  • Examiners award marks for correct endpoints and shading

Representing Inequalities on Number Lines

x > a

  • Open circle at a

  • Shade to RIGHT

x ≥ a

  • Closed circle at a

  • Shade to RIGHT

x < a

  • Open circle at a

  • Shade to LEFT

x ≤ a

  • Closed circle at a

  • Shade to LEFT

a < x < b

  • Two open circles

  • Shade between a and b

a ≤ x ≤ b

  • Two closed circles

  • Shade between a and b

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Inequalities From Word Problems (Most Common Exam Source)

  • Translate English statements into inequality symbols

  • Key triggers

    • “At least” → ≥

    • “At most” → ≤

    • “No more than” → ≤

    • “More than” → >

    • “Less than” → <

    • “Minimum” → ≥

    • “Maximum” → ≤

Examples:

  • “Age must be at least 16” → x ≥ 16

  • “Weight must be less than 50 kg” → x < 50

Solving Inequalities Involving Brackets

  • Expand brackets first

    • a(bx + c) → abx + ac

  • Then solve normally

  • Avoid dividing prematurely

  • Keep inequality direction consistent

  • Always simplify before number line representation

Example:

  • 4(3 − x) > 12

    • 12 − 4x > 12

    • −4x > 0

    • x < 0

(Sign flips when dividing by negative)

Fractional Inequalities Technique

  • Clear denominator by multiplying both sides by positive value

  • If denominator negative → flip sign

  • Use cross-multiplying with CAUTION

  • Always check domain restrictions

Example:

  • (x + 3)/5 ≥ 2

    • x + 3 ≥ 10

    • x ≥ 7

Absolute Value Inequality Patterns

  • |x| < a → −a < x < a

  • |x| > a → x < −a or x > a

  • Use number line definition

  • Absolute inequalities rarely asked but sometimes appear in extended papers

  • Break into two separate inequalities

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Simultaneous Inequalities

  • Given two conditions

  • Solution = OVERLAP region

  • Use shading logic

    • Shade both individual ranges

    • Final answer = intersection

  • If no overlap → no solution

Example:

  • x > 2 AND x ≤ 8 → 2 < x ≤ 8

  • x < −1 AND x ≥ 5 → no solution

Graphical Inequalities (Coordinate Grid)

  • Only required in linear form

  • Convert inequality into boundary line

    • Replace inequality with equality

  • Use correct boundary type

    • Solid line for ≤ or ≥

    • Dashed line for < or >

  • Shade correct region

    • Test point method

    • Substitute (0,0) unless line passes through origin

  • Identify half-plane above or below line

Example:

  • y > 2x + 1

    • Draw dashed line

    • Test (0,0): 0 > 1 (false)

    • Shade OPPOSITE side of origin

Using Test Points to Verify Regions

  • Substitute easy coordinate

  • If inequality holds → shade that side

  • If not → shade opposite side

  • Used when shading gets confusing

  • Works for both x > expression and y > expression

Inequalities in Simultaneous Graph Regions

  • Two or more inequalities define feasible region

  • Region usually polygon-shaped

  • Asked in optimization or shading questions

  • Mark boundaries clearly

  • Shade intersection only

  • Regions should be neat and labelled

  • Avoid shading entire grid; shade lightly

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Quadratic Inequalities (Higher Ability 0580/4024)

  • Solve quadratic inequality by:

    • Step 1: Solve quadratic equation roots

    • Step 2: Draw rough parabola

    • Step 3: Identify where y is positive or negative

  • If inequality is > 0 → choose region OUTSIDE roots

  • If inequality is < 0 → choose region BETWEEN roots

Example:

  • x² − 5x + 6 > 0

    • Roots: 2 and 3

    • Parabola opens up

    • 0 region outside

    • x < 2 or x > 3

Transforming Inequality Expressions

  • Move variables to one side

  • Move constants to other

  • Factor when needed

  • Multiply only after checking sign rules

  • Convert compound inequality into two statements

Checking Answers in Inequality Questions

  • Substitute values slightly inside and outside range

  • If inequality holds → solution correct

  • If fails → adjust shading or inequality sign

  • Check boundary inclusion

  • Re-check sign flips involving negative multipliers

Number Line Interpretation in Exams

  • Ensure equal spacing

  • Mark key values clearly

  • Label arrows to indicate continuing range

  • For answer requiring interval notation

    • Use correct inequality format

    • Maintain smallest value on left

Trick Questions Examiners Use to Trap Students

  • Negative multiplier trick

    • x < 5 vs −x < −5

  • Inequality written backwards

    • 8 ≥ 3x − 1

  • Compound inequality using words

    • “More than 2 but at most 7”

  • Hidden constraint in word problem

  • Graph inequality containing flipped axes

Inequalities in Real-World Word Problems

Age restrictions

  • “Not younger than 18” → x ≥ 18

Mass, height constraints

  • “No more than 70 kg” → x ≤ 70

Price and budget restrictions

  • “At least 40 dollars” → x ≥ 40

Speed limits

  • “Speed must be between 60 and 110 km/h” → 60 ≤ x ≤ 110

Exam Grade Bands

  • 80 ≤ marks ≤ 100

Time constraints

  • t > 5 or t < 2 patterns for scheduling questions

Inequalities Combined With Algebra

  • Solve linear expression first

  • Then apply inequality formatting

  • Examples

    • 3x + 8 < 2 − x

    • x term rearranged

    • Solve for x

  • Final answer may require number line representation

Intervals and Set Notation (Higher-Level Presentation)

  • Use inequality or set notation

  • Example

    • x < 5 → (−∞, 5)

    • x ≥ 3 → [3, ∞)

  • IGCSE rarely demands set notation, but helpful for clarity

A Star Solution Techniques

  • Always check sign flips

  • Always rewrite final inequality with smallest value first

  • For compound inequality → maintain full chain

  • For quadratic → sketch parabola quickly

  • For graphical → always test point

  • For number line → use correct open/closed circles

  • For word problems → convert English to math carefully

  • For multi-step → rewrite inequality after each transformation

  • For checking → substitute values from final range

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

Educate A Change focuses on truly bringing change in the educational sector worldwide. All our courses are designed to ensure success, and provide complete preparation without requiring any other external material. Therefore, it takes away the expense of academies, books, past paper compilations and other miscellaneous items. Instead, by just the courses we offer, the student can prepare fully for the best grades in their CAIEs, Edexcel and other high-school exams. The earnings from our course-sales are invested in social projects and growing the free educational activities of Educate A Change.