Solve and Write Interval Notation for The Solution Set Calculator

This guide explains how to solve inequalities and write interval notation for solution sets. Use the calculator on the right to quickly find solutions and verify your work.

What is Interval Notation?

Interval notation is a way to represent a set of real numbers using parentheses and brackets. It's commonly used in algebra and calculus to describe the range of values that satisfy an inequality.

There are four main types of interval notation:

(a, b) - Open interval: includes all numbers between a and b, not including a and b themselves
[a, b] - Closed interval: includes all numbers between a and b, including both endpoints
(a, b] - Half-open interval: includes all numbers between a and b, not including a but including b
[a, b) - Half-open interval: includes all numbers between a and b, including a but not including b

Special cases include:

(-∞, a) - All numbers less than a
(a, ∞) - All numbers greater than a
(-∞, ∞) - All real numbers
∅ or { } - The empty set (no solution)

How to Solve Inequalities

Solving inequalities follows similar steps to solving equations, but with some important differences:

Identify the inequality symbol (≥, ≤, >, <)
Isolate the variable on one side
Perform operations that maintain the inequality
Write the solution in interval notation

Key Rules for Solving Inequalities

Adding or subtracting the same number from both sides maintains the inequality
Multiplying or dividing both sides by a positive number maintains the inequality
Multiplying or dividing both sides by a negative number reverses the inequality symbol
When multiplying or dividing by a variable expression, the direction depends on the sign of the expression

Example Problem

Solve the inequality: 3x - 5 > 10

Add 5 to both sides: 3x > 15
Divide both sides by 3: x > 5
Interval notation: (5, ∞)

Writing Interval Notation

Once you've solved the inequality, follow these steps to write interval notation:

Identify the lower and upper bounds of the solution set
Determine if the endpoints are included or excluded
Use the appropriate brackets or parentheses
Write the interval in order from smallest to largest

Remember: Parentheses ( ) indicate that the endpoint is not included, while brackets [ ] indicate that the endpoint is included.

Compound Inequalities

For compound inequalities (with "and" or "or" statements), you may need to write multiple intervals:

x > 2 and x < 5 becomes (2, 5)
x ≤ 3 or x ≥ 7 becomes (-∞, 3] ∪ [7, ∞)

Common Interval Notation Examples

Inequality	Solution Set	Interval Notation
x > 4	All numbers greater than 4	(4, ∞)
x ≤ 7	All numbers less than or equal to 7	(-∞, 7]
2 < x < 8	Numbers between 2 and 8, not including 2 and 8	(2, 8)
-3 ≤ x ≤ 5	Numbers between -3 and 5, including both endpoints	[-3, 5]
x ≠ 0	All real numbers except 0	(-∞, 0) ∪ (0, ∞)

FAQ

What's the difference between interval notation and set notation?

Interval notation uses brackets and parentheses to represent ranges of numbers, while set notation lists all elements of a set. For example, {1, 2, 3} is set notation, while [1, 3] is interval notation representing the same numbers.

How do I know when to use parentheses or brackets?

Use parentheses ( ) for strict inequalities (>, <) and brackets [ ] for inclusive inequalities (≥, ≤). For example, x > 2 uses (2, ∞) while x ≥ 2 uses [2, ∞).

What does ∪ mean in interval notation?

The ∪ symbol means "union" and indicates that two separate intervals are combined. For example, (-∞, 2) ∪ (5, ∞) means all numbers except those between 2 and 5.

Can interval notation represent complex numbers?

No, interval notation is specifically for real numbers on the number line. Complex numbers are represented differently in mathematics.