Linear Inequalities - Number Line

Why Use a Number Line?
Open and Closed Circles
Representing Basic Inequalities
Reading Solutions from a Number Line
From Inequality to Number Line
Common Mistakes
Practice Exercises
Related Articles

1. Why Use a Number Line?

A number line gives you a visual picture of all the numbers that satisfy an inequality. Instead of just writing $x > 3$ , you can see at a glance which part of the number line is included in the solution.

Key idea: A number line diagram shows the boundary point and the direction of the solution set using circles and shaded regions.

2. Open and Closed Circles

There are two types of circles used on the number line:

Circle	Meaning	Used for	Symbol
Open circle (hollow)	boundary point is not included	$<$ and $>$	$\circ$
Closed circle (filled)	boundary point is included	$\leq$ and $\geq$	$∙$

Remember: Strict inequalities ( $<$ , $>$ ) use an open circle. Non-strict inequalities ( $\leq$ , $\geq$ ) use a filled circle.

3. Representing Basic Inequalities

Example 1: $x > 3$ (strict, greater than)

The solution includes all numbers to the right of 3, but not 3 itself.

Open circle at 3 (3 is not included)
Blue region extends to the right toward $+ \infty$
Interval notation: $(3, \infty)$

Example 2: $x \leq - 2$ (non-strict, less than or equal)

The solution includes all numbers to the left of $- 2$ , including $- 2$ itself.

-2

Filled circle at $- 2$ ( $- 2$ is included)
Blue region extends to the left toward $- \infty$
Interval notation: $(- \infty, - 2]$

Example 3: $x \geq 0$ (non-strict, greater than or equal)

The solution includes zero and all positive numbers.

Filled circle at 0 (0 is included)
Blue region extends to the right
Interval notation: $[0, \infty)$

Example 4: $x < 5$ (strict, less than)

The solution includes all numbers to the left of 5, but not 5 itself.

Open circle at 5 (5 is not included)
Blue region extends to the left toward $- \infty$
Interval notation: $(- \infty, 5)$

4. Reading Solutions from a Number Line

When you see a number line diagram, you can convert it back to an inequality:

Identify the boundary point (the number under the circle)
Check the circle type --- open means strict, filled means non-strict
Check the direction of the shaded region

Shaded region	Circle	Inequality	Interval
right of $a$	open	$x > a$	$(a, \infty)$
right of $a$	filled	$x \geq a$	$[a, \infty)$
left of $a$	open	$x < a$	$(- \infty, a)$
left of $a$	filled	$x \leq a$	$(- \infty, a]$

Tip: Think of the open circle as a "hole" --- the boundary value has been removed from the solution.

5. From Inequality to Number Line

Here is a step-by-step process for drawing any linear inequality on a number line:

Step 1: Solve the inequality to get it in the form

x > a

x < a

x \geq a

, or

x \leq a

. Step 2: Draw a horizontal line and mark the boundary value

a

. Step 3: Draw the correct circle at

a

Open circle for $<$ or $>$
Filled circle for $\leq$ or $\geq$

Step 4: Shade the correct direction:

Shade to the right for $>$ or $\geq$
Shade to the left for $<$ or $\leq$

Worked Example

Solve and represent on a number line: $3 x - 6 > 9$

Solving:

3 x - 6 > 9

3 x > 15

x > 5

Number line: Open circle at 5, shade to the right.

Solution:

x \in (5, \infty)

6. Common Mistakes

Mistake 1: Mixing Up Circle Types

For $x > 3$ : use an open circle (3 is not a solution).

For $x \geq 3$ : use a filled circle (3 is a solution).

Mistake 2: Shading the Wrong Direction

For $x < 5$ : shade to the left (toward smaller numbers).

For $x > 5$ : shade to the right (toward larger numbers).

Mistake 3: Forgetting to Solve First

Always solve the inequality completely before drawing the number line. Do not try to graph $3 x - 6 > 9$ directly --- first simplify to $x > 5$ .

Practice Exercises

Inequalities - Number Line - Draw and read number line diagrams
Inequalities - Intervals - Convert between number lines and intervals
Inequalities - addition and subtraction - Solve and graph simple inequalities

Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
Linear Inequalities - Intervals and Sets - Writing solutions with interval notation
Linear Inequalities - Simple Inequalities - Solving basic inequality types
Linear Inequalities - Rules and Formulas - Quick reference card