Math · Algebra & word problems
Inequalities and Solution Sets
Estimated time includes reading and one quiz. Take the time you need.
Start here: key ideas
- An inequality describes a set of values.
- Open endpoints exclude; closed endpoints include.
- Multiplying or dividing by a negative reverses order.
- Context determines whether whole-number answers make sense.
What you’ll be able to do
- Solve and graph one-variable inequalities.
- Reverse the sign after negative multiplication or division.
- Interpret bounds in context.
Solve like an equation while protecting the order relationship.
Optional review
If you’d like to review the basics, visit Solving Linear Equations. You can start this lesson without completing those first.
A set rather than one answer
An inequality compares quantities with <, >, ≤, or ≥. Its solution set includes every value making it true. The boundary separates solutions from nonsolutions. An open circle excludes the boundary; a closed circle includes it.
Greater than: endpoint excluded
Every value to the right of 3 satisfies x > 3, but 3 itself does not. This textbook uses a parenthesis for an excluded endpoint; an open circle means the same thing.
Greater than: endpoint excluded — OpenStax, via Mathematics LibreTexts. CC BY 4.0. Source image reproduced unchanged.
At least: endpoint included
For x ≥ 3, include 3 and every larger value. This textbook uses a square bracket; a filled circle is another common way to show that the endpoint is included.
At least: endpoint included — OpenStax, via Mathematics LibreTexts. CC BY 4.0. Source image reproduced unchanged.
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