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Multiplying Large Numbers
Multiplying by 11
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The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
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Graphing Linear Inequalities
Properties of Numbers and Definitions
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Relatively Prime Numbers
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
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Writing Linear Equations in Slope-Intercept Form
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Adding and Subtracting Rational Expressions with Unlike Denominators
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Simple Trinomials as Products of Binomials
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Properties of Exponents
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Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
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Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
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Arithmetic with Positive and Negative Numbers
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Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
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Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
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Multiplication by 429
Writing Linear Equations in Point-Slope Form
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Introduction to Fractions
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Quadratic and Power Inequalities

Quadratic Inequalities

A Quadratic Inequality is an inequality that can be written in the form: ax2 + bx + c > 0 where a, b, and c  are real numbers with a 0. (or with > replaced by <, , or )

We will solve quadratic inequlities both algebraically and graphically. Algebraically, we use a method which involves factoring and a sign diagram. The steps are below.


Algebraic Method for Solving Polynomial Inequalities

1. Write the inequality so that a polynomial expression f(x) is on the left side and zero is on the right side in one of these forms:

f(x) > 0 f(x) 0 f(x) < 0 f(x) 0

2. Determine the numbers at which the expression f(x) on the left side equals zero. (That is, solve f(x) = 0.) These numbers are the Critical Numbers.

3. Mark on a number line the intervals determined by all of the critical numbers.

4. Use a Sign Diagram to determine which intervals are in the solution set in the following manner:

(a) Select a in each of the intervals and evaluate f(x) test number at the test number.

(b) If the value of f(x) is positive, then f(x) > 0 for all numbers x in the interval.

(c) If the value of f(x) is negative, then f(x) < 0 for all numbers x in the interval.

5. Give the answer either in inequality form or in interval notation.

Now let's try the same problem graphically. Again the first step is to rewrite the inequality so that zero is on the right. Then we graph Y1 = f(x). We then locate the zeros of f(x), that is, the x-intercepts.

The solution to the inequality involves an interval or intervals on the x-axis where the function values are greater than or equal to 0. Where does that happen on this graph?

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