1. Introduction
Graphical representation means showing the solution of an inequality on a number line. This helps us see the range of possible values quickly.
A number line is the simplest visual tool for understanding inequalities and solution sets.
2. Number Line Basics
A number line has points placed at equal distances. Zero is in the middle. Positive numbers are to the right, and negative numbers are to the left.
We use the number line to show which values satisfy an inequality.
2.1. Open Circle
Used for strict inequalities (< or >).
An open circle shows the boundary value is not included.
Example: x > 3 → open circle on 3.
2.2. Closed Circle
Used for non-strict inequalities (≤ or ≥).
A closed circle shows the boundary value is included.
Example: x ≥ 2 → closed circle on 2.
3. Representing Inequalities on a Number Line
We use circles for boundaries and arrows to show the direction of valid values.
3.1. 1. For x > a
Open circle at a, arrow to the right.
3.2. 2. For x ≥ a
Closed circle at a, arrow to the right.
3.3. 3. For x < a
Open circle at a, arrow to the left.
3.4. 4. For x ≤ a
Closed circle at a, arrow to the left.
4. Graphing Intervals
Inequality solutions can also be expressed as intervals:
- (a, b) → open at both ends
- [a, b] → closed at both ends
- (a, b] → open at left, closed at right
- [a, b) → closed at left, open at right
5. Examples
- x > 4 → open circle at 4, arrow to the right.
- x ≤ −2 → closed circle at −2, arrow to the left.
- 1 < x ≤ 5 → open circle at 1, closed circle at 5, line in between.
- x ≥ 0 → closed circle at 0, arrow to the right.
6. Common Mistakes
- Using a closed circle when the inequality is strict (< or >).
- Using an open circle when equality must be included.
- Forgetting the arrow to show direction.
- Reversing the arrow direction by mistake.
- Misreading the number line when negative numbers are involved.
7. Quick Practice
Draw number line representations for:
- x ≥ 7
- x < −1
- −3 < x ≤ 2
- x > 0
8. Summary
- Solutions of inequalities can be shown clearly on a number line.
- Open circle → boundary not included.
- Closed circle → boundary included.
- Arrows show direction of valid values.
- Intervals can represent solution ranges easily.