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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Quadratic Inequalities

A quadratic inequality has the form ax + bx + c > 0 (or <, or , or ). The highest exponent is 2. The next few examples show how to solve quadratic inequalities.

Solving Quadratic Inequalities


Solve the quadratic inequality x - x < 12.


Write the inequality with 0 on one side, as x - x - 12 < 0. This inequality is solved with values of x that make x - x - 12 negative (< 0). The quantity x - x - 12 changes from positive to negative or from negative to positive at the points where it equals 0. For this reason, first solve the equation x - x - 12 = 0.

x - x - 12 = 0

(x - 4)(x + 3) = 0

x = 4 or x = -3

Locating -3 and 4 on a number line, as shown in Figure 3, determines three intervals A, B, and C.

Decide which intervals include numbers that make x - x - 12 negative by substituting any number from each interval in the polynomial. For example,

choose -4 from interval A: (-4) - (-4) - 12 = 8 > 0;

choose 0 from interval B: 0 - 0 - 12 = -12 < 0;

choose 5 from interval C: 5 - 5 - 12 = 8 > 0.

Only numbers in interval B satisfy the given inequality, so the solution is (-3, 4). A graph of this solution is shown in Figure 4.

Solving Polynomial Inequalities


Solve the inequality x(x-1)(x+3) 0.


This is not a quadratic inequality. If the three factors are multiplied, the highest-degree term is x. However, it can be solved in the same way as a quadratic inequality because it is in factored form. First solve the corresponding equation.

x(x - 1)(x + 3) = 0

x = 0 or x - 1 = 0 or x + 3 = 0

x = 0 or x = 1 or x = -3

These three solutions determine four intervals on the number line: , (-3, 0), (0, 1) and . Substitute a number from each interval into the original inequality to determine that the solution includes the numbers less than or equal to -3 and the numbers that are equal to or between 0 and 1. See Figure 5.

In interval notation, the solution is

* The symbol indicates the union of two sets, which includes all elements in either set.

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