Inequalities

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Represent and interpret inequalities, including on a number line
  2. Construct, solve and interpret linear inequalities
  3. Represent and interpret linear inequalities in two variables graphically
  4. List inequalities that define a given region

1. Understanding Inequalities

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What is an inequality?

An inequality is a mathematical statement that compares two values and shows that one is greater than or less than the other. Unlike an equation (which uses =), an inequality tells us that the two sides are not equal.

Inequality symbols

There are four main inequality symbols you need to know:

  • < means "less than"
    Example: x < 5 means "x is less than 5"

  • > means "greater than"
    Example: x > 3 means "x is greater than 3"

  • ≤ means "less than or equal to"
    Example: x ≤ 10 means "x is less than or equal to 10"

  • ≥ means "greater than or equal to"
    Example: x ≥ 2 means "x is greater than or equal to 2"

Strict inequalities vs inclusive inequalities

Strict inequalities use < or > symbols. These do NOT include the boundary value.

  • If x > 5, then x could be 6, 7, 8, 9, ... but NOT 5 itself
  • The word "strict" means we are strict about not including the endpoint

Inclusive inequalities use ≤ or ≥ symbols. These DO include the boundary value.

  • If x ≥ 5, then x could be 5, 6, 7, 8, 9, ...
  • Notice that 5 is now included in the possible values

2. Representing Inequalities on a Number Line

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How to draw inequalities on a number line

When showing an inequality on a number line, you need to mark the boundary value and show which direction the allowed values go.

Key rules for number lines:

  1. Open circle (○) — use this for strict inequalities (< or >)
    This shows the endpoint is NOT included

  2. Closed circle (●) — use this for inclusive inequalities (≤ or ≥)
    This shows the endpoint IS included

  3. Line or arrow — connect the circles or draw an arrow to show all the values that satisfy the inequality

Examples of number line representations

Example 1: x > 2

  • Draw an open circle at 2 (because > is strict, so 2 is not included)
  • Draw an arrow pointing to the right (towards larger numbers)

Example 2: x ≤ -1

  • Draw a closed circle at -1 (because ≤ is inclusive, so -1 is included)
  • Draw an arrow pointing to the left (towards smaller numbers)

Example 3: -3 ≤ x < 1

This is a double inequality because it has two boundaries.

  • Draw a closed circle at -3 (because ≤ includes -3)
  • Draw an open circle at 1 (because < does not include 1)
  • Draw a line connecting the two circles
  • This shows that x can be any value from -3 up to (but not including) 1

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