Solution Sets

Solution Sets for Equations

The set containing all the solutions of an equation is called the solution set for that equation.

If an equation has no solutions, we write for the solution set. means the null set (or empty set).

Equation
Solution Set
3x + 5 = 11
{2}
x2 = x
{0, 1}
x + 1 = 1 + x
R (the set of all real numbers)
x + 1 = x
(the empty set)

Solution Sets for Inequalities

Solution sets for inequalities are often infinite sets; we can't list all the numbers. So, we use a special notation.

Example:

Solve the inequality

x + 2 –3.

By subtracting 2 from both sides, we get the equivalent inequality

x –5.

So, the solution set is

{x | x –5}.

(To read this, you would say: "x such that x is greater than negative five." The | symbol means "such that" in this case.)

Often, the solutions to inequalities are also written in interval notation.