Solution Set Formula

Solution set is the complete set of all values that satisfy a given equation or inequality — it may be empty, finite, or infinite.

The Formula

S={x∣f(x)=0}S = \{x \mid f(x) = 0\}

When to use: Not just one answer, but ALL answers that work — an inequality like x>3x > 3 has infinitely many.

Quick Example

x2=4x^2 = 4 has solution set {−2,2}\{-2, 2\}. x>3x > 3 has solution set (3,∞)(3, \infty).

Notation

Set notation {…}\{\ldots\} for discrete solutions, interval notation (a,b)(a, b), [a,b][a, b] for continuous ranges. ∅\emptyset or {}\{\} for no solution.

What This Formula Means

The complete set of all values that satisfy a given equation or inequality — it may be empty, finite, or infinite.

Not just one answer, but ALL answers that work — an inequality like x>3x > 3 has infinitely many.

Formal View

The solution set of f(x)=g(x)f(x) = g(x) over domain DD is S={x∈D∣f(x)=g(x)}S = \{x \in D \mid f(x) = g(x)\}. Cases: S=∅S = \emptyset (no solution), ∣S∣=1|S| = 1 (unique), ∣S∣=n|S| = n (finite), or ∣S∣=∣R∣|S| = |\mathbb{R}| (identity).

Worked Examples

Example 1

easy
What is the solution set of x2=25x^2 = 25?

Answer

{−5,5}\{-5, 5\}

First step

1
Find all values where x2=25x^2 = 25: x=5x = 5 or x=−5x = -5.

Full solution

  1. 2
    Write as a set: {5,−5}\{5, -5\}.
  2. 3
    The solution set contains every value that satisfies the equation.
A solution set is the collection of all values that make the equation true. It can contain zero, one, two, or infinitely many elements.

Example 2

medium
What is the solution set of x+3>5x + 3 > 5?

Example 3

medium
Find the solution set of x2−7x+12=0x^2-7x+12=0.

Common Mistakes

  • Giving one value when there are many - inequalities and quadratics often have whole ranges or multiple answers.
  • Forgetting the empty set is a valid answer - if nothing satisfies it, write ∅\emptyset.
  • Mixing up open and closed brackets in interval notation - (( excludes the endpoint, [[ includes it.

Why This Formula Matters

Many statements don't have a single answer: inequalities have ranges, some equations have none, quadratics have two. Naming the whole set — with set or interval notation — keeps you from reporting one answer when the truth is a range or nothing at all. Recognizing it by "Am I describing EVERY value that satisfies the statement, not just one?" — rather than by familiar numbers — is what lets a student tell it apart from solution (single) and interval notation and empty set in a mixed problem set.

Frequently Asked Questions

What is the Solution Set formula?

The complete set of all values that satisfy a given equation or inequality — it may be empty, finite, or infinite.

How do you use the Solution Set formula?

Not just one answer, but ALL answers that work — an inequality like x>3x > 3 has infinitely many.

What do the symbols mean in the Solution Set formula?

Set notation {…}\{\ldots\} for discrete solutions, interval notation (a,b)(a, b), [a,b][a, b] for continuous ranges. ∅\emptyset or {}\{\} for no solution.

Why is the Solution Set formula important in Math?

Many statements don't have a single answer: inequalities have ranges, some equations have none, quadratics have two. Naming the whole set — with set or interval notation — keeps you from reporting one answer when the truth is a range or nothing at all. Recognizing it by "Am I describing EVERY value that satisfies the statement, not just one?" — rather than by familiar numbers — is what lets a student tell it apart from solution (single) and interval notation and empty set in a mixed problem set.

What do students get wrong about Solution Set?

The procedure for solution set is the easy part; the trap is giving one value when there are many. Asking "Am I describing EVERY value that satisfies the statement, not just one?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Solution Set formula?

Before studying the Solution Set formula, you should understand: solution concept.

← Back to Solution Set See All Examples Practice Problems