Solution Set — Definition, Formula & Examples
A solution set is the set of all values that make an equation, inequality, or system true. When you solve a math problem, the solution set collects every answer into one group using set notation.
Given an equation or inequality in one or more variables, the solution set is the set of all ordered tuples from the replacement domain that, when substituted, produce a true statement. If no value satisfies the condition, the solution set is the empty set .
How It Works
To find a solution set, solve the equation or inequality as you normally would, then express every valid answer inside curly braces. For a single-variable equation like , the solution set is . For a quadratic like , there are two solutions, so the solution set is . An inequality such as has infinitely many solutions, written in set-builder notation as . If no value works—like —the solution set is .
Worked Example
Problem: Find the solution set of the equation .
Factor: Factor the quadratic expression.
Apply the Zero-Product Property: Set each factor equal to zero and solve.
Write the solution set: Collect both solutions into set notation.
Answer: The solution set is .
Why It Matters
Solution sets appear throughout Algebra 1, Algebra 2, and linear algebra whenever you need to describe all answers at once. In data science and engineering, defining the full solution set of a system of equations tells you exactly what configurations are possible, rather than just one particular answer.
Common Mistakes
Mistake: Listing only one solution when multiple exist, for example writing instead of for .
Correction: Always check whether the equation has more than one solution. Quadratics can have two, and inequalities often have infinitely many. The solution set must include every valid value.