What Is a Compound Inequality?
Wondering what a compound inequality is? This guide explains compound inequalities clearly, covers “and” and “or” types, and shows you how to solve and graph them step by step.
If you are studying algebra and ran into the term compound inequality, you are in the right place. A compound inequality is two inequalities joined together into a single statement. Instead of saying a value is greater than one number, you are saying it falls within a range or satisfies one of two conditions at the same time. Once you see how they work, they are not complicated. This guide breaks down what a compound inequality is, the two types you will encounter, and how to solve them.
The Basic Definition
A compound inequality combines two separate inequalities using the word and or the word or.
A simple inequality looks like this: x > 3
A compound inequality looks like this: x > 3 and x < 10
That second example says x must be greater than 3 AND less than 10 at the same time. Only values between 3 and 10 satisfy both conditions.
You will also see compound inequalities written in a condensed form: 3 < x < 10
This means the same thing. X sits between 3 and 10.
The Two Types of Compound Inequalities
“And” Compound Inequalities (Intersection)
An “and” compound inequality requires both conditions to be true at the same time. The solution is the set of values that satisfies both inequalities.
Example: x > 2 and x < 8
The solution is all values of x between 2 and 8. On a number line, you shade the region between 2 and 8 with open circles at both endpoints (since 2 and 8 themselves are not included).
If the inequality uses greater than or equal to (≥) or less than or equal to (≤), you use closed circles at the endpoints instead.
Real-world example: A theme park ride requires riders to be at least 48 inches tall and no more than 78 inches tall. That is an “and” compound inequality: height ≥ 48 and height ≤ 78, or written as 48 ≤ height ≤ 78.
“Or” Compound Inequalities (Union)
An “or” compound inequality requires only one of the two conditions to be true. The solution is the set of values that satisfies either inequality.
Example: x < 1 or x > 6
The solution includes everything less than 1 and everything greater than 6. On a number line, you shade two separate regions pointing outward in both directions.
“Or” inequalities tend to produce solutions that go off in two directions rather than staying in a contained range.
Real-world example: A store offers a discount to customers under 12 years old or over 65. That is an “or” compound inequality: age < 12 or age > 65.
How to Solve a Compound Inequality
Solving a compound inequality follows the same rules as solving a regular inequality, applied to both parts.
Example: Solve 2 < 3x + 1 < 13
Treat both sides the same way:
- Subtract 1 from all three parts: 2 – 1 < 3x < 13 – 1 → 1 < 3x < 12
- Divide all three parts by 3: 1/3 < x < 4
The solution is all values of x between 1/3 and 4.
One important rule: if you multiply or divide all parts by a negative number, flip both inequality signs.
How to Graph a Compound Inequality
Graphing on a number line uses two simple rules:
- Open circle at a value means that number is not included (< or >)
- Closed circle means the number is included (≤ or ≥)
For “and” inequalities, shade the region between the two values. For “or” inequalities, shade outward from both values in opposite directions.
Quick Reference
| Type | Connector | Solution Set | Graph |
|---|---|---|---|
| And | and / between | Values in the middle | Shade between two points |
| Or | or | Values outside | Shade outward from two points |
The Short Answer
A compound inequality is two inequalities combined with “and” or “or.” An “and” compound inequality finds values that satisfy both conditions at once, producing a range. An “or” compound inequality finds values that satisfy either condition, producing two separate regions. Solve them the same way you solve regular inequalities, just apply each step to all parts of the statement.
