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:

Inequality	Symbol	Circle Type	Arrow 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!

┌─────────────────────────────────────────────────────────────┐
│         GRAPHICAL REPRESENTATION - NUMBER LINE GUIDE         │
└─────────────────────────────────────────────────────────────┘

BASIC INEQUALITIES ON NUMBER LINE:

    x > 3:
    
    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5
    
         Empty circle at 3, arrow to the right


    x ≥ 3:
    
    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5
    
         Filled circle at 3, arrow to the right


    x < 2:
    
    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5
    
         Empty circle at 2, arrow to the left


    x ≤ 2:
    
    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5
    
         Filled circle at 2, arrow to the left


COMPOUND INEQUALITY (AND) — "x > 2 and x < 5":

    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5   6
    
         Open circles at 2 and 5, shaded in between
         This means 2 < x < 5


COMPOUND INEQUALITY (OR) — "x < 1 or x > 4":

    <───┼───┼───┼───┼───┼───┼───►
        0   1   2   3   4   5
    
         Arrows pointing outward (both left and right)


SUMMARY TABLE:

┌───────────────────┬────────────────┬───────────────────────┐
│   Inequality      │   Circle Type   │     Arrow Direction   │
├───────────────────┼────────────────┼───────────────────────┤
│     x > a         │    Open (○)     │     Right (→)         │
│     x < a         │    Open (○)     │     Left (←)          │
│     x ≥ a         │   Closed (●)    │     Right (→)         │
│     x ≤ a         │   Closed (●)    │     Left (←)          │
│   a < x < b       │   Open (○) at   │     Between a and b   │
│                   │   both ends     │                       │
│   a ≤ x ≤ b       │   Closed (●)    │     Between a and b   │
│                   │   at both ends  │                       │
└───────────────────┴────────────────┴───────────────────────┘

Solved Examples

Worked Example

Represent the solution x > 3 on a number line.

Solution

Step 1: Locate 3 on the number line
Step 2: Since it's x > 3 (strictly greater), use an open circle at 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

Worked Example

Solve and graph on a number line: 2x + 1 ≤ 9

Solution

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

Answer: Closed circle at 4, arrow pointing left

Worked Example

Solve and graph: −3 ≤ 2x − 1 < 5, where x ∈ integers (x is an integer)

Solution

Step 1: Solve as compound inequality: −3 ≤ 2x − 1 and 2x − 1 < 5
Step 2: First: −3 ≤ 2x − 1 → −2 ≤ 2x → −1 ≤ x
Step 3: Second: 2x − 1 < 5 → 2x < 6 → x < 3
Step 4: Combined: −1 ≤ x < 3
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