Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Interval Notation and Graphs

If an inequality involves a variable, then which real numbers can be used in place of the variable to obtain a correct statement? The set of all such numbers is the solution set to the inequality. For example, x < 3 is correct if x is replaced by any number that lies to the left of 3 on the number line:

1.5 < 3,    0 < 3,     and    -2 < 3

The set of real numbers to the left of 3 is written in set notation as {x | x < 3}, in interval notation as (-∞, 3), and graphed below:

Note that -∞ (negative infinity) is not a number, but it indicates that there is no end to the real numbers less than 3. The parenthesis used next to the 3 in the interval notation and on the graph means that 3 is not included in the solution set to x < 3.

An inequality such as x ≥ 1 is satisfied by 1 and any real number that lies to the right of 1 on the number line. The solution set to x ≥ 1 is written in set notation as {x | x ≥ 1}, in interval notation as [1, ∞), and graphed as below:

The bracket used next to the 1 in the interval notation and on the graph means that 1 is in the solution set to x ≥ 1.

The solution set to an inequality can be stated symbolically with set notation and interval notation, or visually with a graph. Interval notation is popular because it is simpler to write than set notation. The interval notation and graph for each of the four basic inequalities is summarized as follows.


Basic Interval Notation (k any real number)

Inequality Solution Set with Interval Notation Graph
x > k (k, )
x ≥ k [k, )
x < k (-, k)
x ≤ k (-, k]



Interval notation and graphs

Write the solution set to each inequality in interval notation and graph it.

a) x > -5

b) x ≤ 2


a) The solution set to the inequality x > -5 is {x | x > -5}. The solution set is the interval of all numbers to the right of -5 on the number line. This set is written in interval notation as (-5, ), and it is graphed as follows:

b) The solution set to x ≤ 2 is {x | x ≤ 2}. This set includes 2 and all real numbers to the left of 2. Because 2 is included, we use bracket at 2. The interval notation for this set is (-∞, 2]. The graph is shown below.

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