Glossary
Solution set
Also written: solution · solutions · solution sets
Definition
The solution set is every value that makes an equation or inequality true. An equation often has one solution, an inequality usually has infinitely many, and some have none at all.
How big a solution set can be
| Sentence | Solution set |
|---|---|
| one value, | |
| every number below | |
| two values, and | |
| empty — no number works | |
| every real number |
Empty is a real answer
“No solution” means the solution set is empty, and that is a complete answer rather than a failure to finish. It happens whenever the variable cancels and leaves a false statement, like — see variables on both sides.
Checking membership
To test whether a number belongs, substitute it into the original sentence and see whether the result is true. This is the fastest way to catch a sign error, and it works even when you are unsure of your method.
Lessons that use this term
- How to Solve One-Step Inequalities
Solve one-step inequalities the same way as equations, plus the one rule that changes: why the sign flips when you multiply or divide by a negative.
- Solving Equations with Variables on Both Sides
Solve equations with variables on both sides by collecting variable terms on one side, plus how to recognize no solution and infinitely many solutions.
- Solution Sets of Linear Systems
Read the shape of a solution set from the pivots: none, exactly one, or infinitely many, written in parametric vector form as a particular solution plus directions.