Linear Inequations • Topic 2 of 2

Graphical Representation on Number Line

How to represent inequations on a number line? A number line is used to visually represent the solution set of an inequation. Different types of endpoints indicate whether the endpoint is included or excluded.

Rules for number line representation:

InequalitySymbolCircle TypeArrow Direction
x > a>Open circle (○)Points to the right
x < a<Open circle (○)Points to the left
x ≥ a≥Closed circle (●)Points to the right
x ≤ a≤Closed circle (●)Points to the left

Types of solution sets:

  • Single inequality: One arrow on the number line
  • Compound inequality (and): Overlapping region between two values
  • Compound inequality (or): Two separate arrows going outward

Real-life analogy: The number line is like a ruler marking all possible numbers. An open circle is like a "do not touch" sign — the exact point is not included. A closed circle is like a "touch here" sign — the point is included!

Four number lines: x > 3 open ray right, x ≥ 3 filled ray right, x < 2 open ray left, x ≤ 2 filled ray left Open circle for < >, filled circle for ≤ ≥ x > 3012345x ≥ 3012345x < 2012345x ≤ 2012345 A bounded solution set negative 2 ≤ x < 3: filled circle at negative 2, open circle at 3, segment shaded between A bounded set: −2 ≤ x < 3 −3−2−101234 −2 included (≤) 3 excluded (<) Integers in the set: −2, −1, 0, 1, 2
Solved Examples
1
Worked Example
Represent the solution x > 3 on a number line.
Solution
  1. Step 1: Locate 3 on the number line
  2. Step 2: Since it's x > 3 (strictly greater), use an open circle at 3
  3. Step 3: Draw an arrow pointing to the right (all numbers greater than 3)

Answer: Open circle at 3 with arrow to the right

2
Worked Example
Solve and graph on a number line: 2x + 1 ≤ 9
Solution
  1. Step 1: Solve: 2x + 1 ≤ 9 → 2x ≤ 8 → x ≤ 4
  2. Step 2: Locate 4 on the number line
  3. Step 3: Since it's x ≤ 4 (less than or equal), use a closed (filled) circle at 4
  4. Step 4: Draw an arrow pointing to the left

Answer: Closed circle at 4, arrow pointing left

3
Worked Example
Solve and graph: −3 ≤ 2x − 1 < 5, where x ∈ integers (x is an integer)
Solution
  1. Step 1: Solve as compound inequality: −3 ≤ 2x − 1 and 2x − 1 < 5
  2. Step 2: First: −3 ≤ 2x − 1 → −2 ≤ 2x → −1 ≤ x
  3. Step 3: Second: 2x − 1 < 5 → 2x < 6 → x < 3
  4. Step 4: Combined: −1 ≤ x < 3
  5. Step 5: Since x ∈ integers, x = −1, 0, 1, 2

Answer: On number line: closed circle at −1, open circle at 3, all integers between shaded

Key Points

  • Open circle (○) for < and > (endpoint not included)
  • Closed circle (●) for ≤ and ≥ (endpoint included)
  • Arrow to the right for > and ≥ (greater than)
  • Arrow to the left for < and ≤ (less than)
  • Compound "and" inequalities: shaded between two points
  • Compound "or" inequalities: two separate arrows outward
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.On a number line, $x>3$ is shown by:
Explanation: Open circle (excluded) + arrow right.
Q2.A closed (filled) circle indicates the endpoint is:
Explanation: Included endpoint.
Q3.An open circle indicates the endpoint is:
Explanation: Excluded endpoint.
Q4.$x\le5$ is shown with:
Explanation: Closed at $5$, arrow left.
Practice more MCQs → 📝 Assignment · 35 marks