x < 4
x can be any number smaller than 4, but 4 itself is not included.
Free Algebra Guide
Inequalities and Interval Notation
Learn how to graph inequalities on a number line, choose open or closed endpoints, and translate between inequality notation and interval notation.
An inequality describes a range of possible values rather than one single number. The same solution set can be written with inequality symbols, shown on a number line, or written using interval notation.
Once you understand boundary values, endpoint inclusion, and the direction of the solution set, moving between these representations becomes much easier.
On This Page
- Inequality symbols
- Graphing inequalities
- Open vs. closed endpoints
- Which direction to shade
- Interval notation basics
- Brackets vs. parentheses
- Infinity in interval notation
- Conversion chart
- Compound inequalities
- “Or” inequalities and unions
- All real numbers and no solution
- Worked examples
- Common mistakes
- Reliable strategy
- Interactive practice
What Do the Inequality Symbols Mean?
The inequality symbol tells you whether values smaller than, greater than, or equal to a boundary value belong to the solution set.
x > 4
x can be any number larger than 4, but 4 itself is not included.
x ≤ 4
Values below 4 are included, and 4 is included too.
x ≥ 4
Values above 4 are included, and the boundary value 4 is included too.
The words
“or equal to”
determine whether the boundary value itself
belongs to the solution.
How to Graph an Inequality on a Number Line
A one-variable inequality is usually graphed using an endpoint at the boundary value and a shaded region showing all values that satisfy the inequality.
1. Find the boundary
Locate the number being compared with x.
2. Choose the endpoint
Decide whether the boundary is included
or excluded.
3. Shade the solution
Shade left for smaller values and right
for larger values.
Example: x < 3
3
The circle at 3 is open because 3 is not included. The shading extends left because the inequality contains values less than 3.
Open vs. Closed Circles
The endpoint tells you whether the exact boundary value is part of the solution.
Endpoint Is Excluded
< or >
Use an open circle when the inequality is strict.
Endpoint Is Included
≤ or ≥
Use a closed circle when equality is allowed.
Quick rule:
No equality line under the symbol =
open.
Equality line under the symbol =
closed.
Which Direction Should You Shade?
Numbers become smaller as you move left on a number line and larger as you move right.
| Inequality | Meaning | Shade |
|---|---|---|
| x < a | Values less than a | Left |
| x ≤ a | Values less than or equal to a | Left |
| x > a | Values greater than a | Right |
| x ≥ a | Values greater than or equal to a | Right |
Less than ← left.
Greater than → right.
What Is Interval Notation?
Interval notation describes the same collection of numbers using endpoints and parentheses or brackets instead of inequality symbols.
x ≥ 4
↔
[4, ∞)
The interval begins at 4 and continues forever to the right.
−2 < x ≤ 5
↔
(−2, 5]
The left endpoint is excluded, while the right endpoint is included.
Brackets vs. Parentheses
(3, 7)
A parenthesis means the endpoint is not included.
[3, 7]
A bracket means the finite endpoint is included.
| Inequality Symbol | Number-Line Endpoint | Interval Symbol |
|---|---|---|
| < or > | Open circle | Parenthesis ( ) |
| ≤ or ≥ | Closed circle | Bracket [ ] |
Infinity in Interval Notation
Infinity is not a number that can be reached. It describes a direction that continues without end.
Infinity Always Uses Parentheses
Never place a bracket next to ∞ or −∞. Infinity cannot be included as an endpoint.
x < 4
→
(−∞, 4)
x ≥ −2
→
[−2, ∞)
Inequality to Interval Notation Conversion Chart
| Inequality | Interval Notation | Graph |
|---|---|---|
| x < a | (−∞, a) | Open at a, shade left |
| x ≤ a | (−∞, a] | Closed at a, shade left |
| x > a | (a, ∞) | Open at a, shade right |
| x ≥ a | [a, ∞) | Closed at a, shade right |
| a < x < b | (a, b) | Open at both; shade between |
| a ≤ x ≤ b | [a, b] | Closed at both; shade between |
| a < x ≤ b | (a, b] | Open left, closed right |
| a ≤ x < b | [a, b) | Closed left, open right |
Compound Inequalities with “And”
An “and” inequality usually describes values lying between two boundaries.
−3 ≤ x < 5
−3
5
[−3, 5)
The solution contains the numbers from −3 to 5. The −3 is included, while 5 is excluded.
AND usually means BETWEEN.
“Or” Inequalities and Union Notation
An “or” inequality can create two separate portions of the number line.
x < −2
or
x ≥ 4
The interval notation is:
(−∞, −2) ∪ [4, ∞)
The symbol ∪ means union. It joins the two intervals into one solution set.
Think of union as “combine these solution regions.”
When an “or” inequality points outward, the graph usually shades the two outside regions.
All Real Numbers and No Solution
(−∞, ∞)
Every real number belongs to the solution set.
On the number line, the entire line is shaded.
∅
No real number satisfies the condition.
This is also described as no solution.
How Do You Type ∅?
In the interactive practice tool, you do not need a special keyboard symbol. You can type empty set or no solution when the solution is empty.
Worked Inequality and Interval Notation Examples
x > 6
- Boundary value: 6.
- > excludes 6, so use a parenthesis.
- Values extend toward positive infinity.
- Answer: (6, ∞).
(−∞, 2]
- The finite boundary is 2.
- The bracket includes 2.
- The interval extends left.
- Answer: x ≤ 2.
−4 < x ≤ 7
- Left boundary −4 is excluded.
- Right boundary 7 is included.
- Shade between the boundaries.
- Answer: (−4, 7].
x ≤ −1 or x > 5
- The left interval ends at −1 and includes it.
- The right interval begins at 5 but excludes it.
- Join the intervals with ∪.
- Answer: (−∞, −1] ∪ (5, ∞) .
Common Inequality and Interval Notation Mistakes
Mistake 1: Using a Closed Circle for < or >
Strict inequalities exclude the boundary, so the endpoint must be open.
Mistake 2: Using an Open Circle for ≤ or ≥
“Or equal to” includes the boundary, so use a closed endpoint.
Mistake 3: Shading in the Wrong Direction
Less-than solutions extend left. Greater-than solutions extend right.
Mistake 4: Using a Bracket with Infinity
Infinity is never an included endpoint, so always use a parenthesis next to ±∞.
Mistake 5: Reversing the Interval Endpoints
Interval notation is always written from the smaller value on the left to the larger value on the right.
Mistake 6: Confusing “And” with “Or”
“And” usually represents the overlapping region between boundaries. “Or” combines separate valid regions.
Mistake 7: Treating ∅ Like the Number Zero
∅ means the solution set contains no elements. It does not mean the solution is x = 0.
A Reliable Strategy for Inequality and Interval Problems
1. Identify the boundary values
Find the finite numbers that separate
included and excluded regions.
2. Decide whether each boundary is included
< and > mean excluded.
≤ and ≥ mean included.
3. Determine the solution region
Decide whether the solution goes left,
right, between the endpoints, or outside
them.
4. Translate the endpoint style
Open circle ↔ parenthesis.
Closed circle ↔ bracket.
5. Handle infinity correctly
Always use parentheses at ±∞.
6. Check the final set
Ask whether your inequality, interval,
and graph all describe exactly the same
numbers.
The key question is:
which values belong to the solution set?
Practice What You Learned
Try the Inequalities & Interval Notation Interactive Tool
Practice graphing inequalities directly on a number line, reading number-line graphs, and converting back and forth between inequality notation and interval notation.
Start Inequality & Interval PracticeRelated Algebra Resources
Inequalities build on many of the same ideas used in equations, coordinate graphs, slope, and linear functions.
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