Key Terms

Inequality: A statement that one expression is greater or less than another
Integer solution: A whole number that satisfies the inequality
Number line: Visual representation — open circle (strict) or closed circle (inclusive)
Compound inequality: Two inequalities combined (e.g. −1 < x ≤ 3)
Strict inequality: < or > (does NOT include the end value)
Inclusive inequality: ≤ or ≥ (DOES include the end value)

Must-Know Facts

Solve inequalities like equations EXCEPT: flip the sign when multiplying or dividing by a NEGATIVE
Open circle on number line = strict inequality (< or >)
Closed (filled) circle = inclusive inequality (≤ or ≥)
For integer solutions of −2 < x ≤ 3: list −1, 0, 1, 2, 3 (NOT −2)
x > 3 on a number line: arrow pointing right from open circle at 3
Inequalities can have infinitely many solutions (a range, not one value)

Key Symbols

< means "less than" (strictly)
> means "greater than" (strictly)
≤ means "less than or equal to"
≥ means "greater than or equal to"
FLIP the inequality sign when × or ÷ by a negative number

Common Mistakes

Not flipping the inequality: When dividing or multiplying both sides by a negative number, reverse the inequality sign — −2x > 6 becomes x < −3
Open vs closed circle: Use an open circle for strict inequalities (< or >) and a filled circle for ≤ or ≥ — mixing these up loses marks
Including the boundary in integer solutions: For −2 < x ≤ 3, the value −2 is NOT included — list only −1, 0, 1, 2, 3
Solving a compound inequality incorrectly: Apply every operation to all three parts — for −1 < 2x + 1 ≤ 7, subtract 1 then divide by 2 throughout

This topic summary covers Knowledge Organiser: Linear Inequalities within Linear Inequalities for GCSE Mathematics. Revise Linear Inequalities in Algebra for GCSE Mathematics with 14 exam-style questions and 11 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 8 of 8 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.

Practice questions for Linear Inequalities

Which of the following correctly describes how to represent x > 3 on a number line?

A. A closed (filled) circle at 3, with an arrow pointing to the right
B. An open (empty) circle at 3, with an arrow pointing to the right
C. An open (empty) circle at 3, with an arrow pointing to the left
D. A closed (filled) circle at 3, with an arrow pointing to the left

1 markfoundation

When solving an inequality, the direction of the inequality sign must reverse if you multiply or divide both sides by a negative number. Explain why this rule is necessary. You may use an example to support your explanation.

2 marksstandard

Quick recall flashcards

Open vs Closed Circle on a number line

Open circle (hollow) for < and > — endpoint NOT included. Closed circle (filled) for ≤ and ≥ — endpoint IS included.

The Flip Rule for inequalities

When you multiply or divide BOTH sides by a NEGATIVE number, you MUST reverse (flip) the inequality sign. Example: -2x > 6 becomes x < -3

Knowledge Organiser: Linear Inequalities